
QassQ. A 5 -0 ? ^ 
Book SjoJa , 



/ 
bobinsojST's mathematicax, series. 



A NEW TEEATISE 



ON THE ELEMENTS OF 



THE DIFFERENTIAL AND INTEGRAL 



CALCULUS. 



EDITED BY 

I. F. QUINBY, A.M., LL.D., 

PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY, UNIVERSITY OF ROCHESTER. 



NEW YORK: 
IVISON, PniNNEY, BLAKKMAX. .'v CO. 

47 ifc 4!) (JUEKNK ."^TKKK l'. 

PHILADELl'HIA: .1. U. l.irriNcWlT .S: VO. 

1 SG8. 



Entered, according to act of Congress, in the year 1867, hy 

DANIEL W. FISH, A.M., 

In the Clerk's Office of tlie District Court of the United States for the Eastern District 

of New York. 



%}P ' 



Electrotyped and Printed by Geo. C. Rand & Avery, Boston. 



PEEFACE. 



The design in preparing this treatise on the Differential and In- 
tegral Calculus has been, not so much to produce a work that should 
cover the whole ground of this extensive and rapidly extending branch 
of mathematics, as to produce one that should be complete within the 
limits assigned it, and adapted to the wants of students in the higher 
schools and colleges of this country. Many of the subjects are much 
more fully discussed in this volume than in other elementary trea- 
tises ; while many are entirely omitted here which are generally 
included in such works, though they are not essential to, and arc 
rarely embraced in, the college course in this or in other countries. 
The necessity devolved on the author, either to be limited in the num- 
ber and full in the treatment of the subjects selected, or full iu the 
number of subjects, and limited iu their discussion. The lornior 
choice was taken, keeping iu view the logical and progressive develop- 
ment of the principles. 

This will account for the omission, among other subjects, oC the 
integration of dilferential ecpiations of iho dilloront />rdors, and uf 
the "Calculus of Variations," the latter of which, ^^ hen I'nlly trcati-il, 
would make a volume etpial to the present in si/.e. 



4 PREFACE. 

It will be found, however, that the time usually given to this study 
will render it impossible to take, in course, all the subjects herein 
treated. The following are what may be left out in the class-room 
without serious breaks in continuity : — 

Differential Calculus. — Part First. — Section V., from Article 
68 to the end of the Section. The whole of Section YII. Section 
X., from Article 110 to the end of the Section. The whole of Sec- 
tion XII. Section XIV., from Article 139 to the end of the Section. 

Differential Calculus. — Part Second. — The whole of Section 
II. Section IV., from Article 177 to Article 181 ; from Article 188 
to the end of the Section. 

Integral Calculus. — From Example 4, Section IV., to the end 
of the Section. The whole of Sections IX. and X. 

It will be observed that the fundamental proposition of the Differ- 
ential Calculus is based on the doctrine of limits ; and that of the 
Integral Calculus, on that of the summation of an infinite series of 
infinitely small terms. The author adopts these methods merely on 
logical grounds, but ventures the opinion that these, and what are 
called the infinitesimal methods, are based on the same metaphysical 

principles. 

THE AUTHOK 

November, 1867. 



CONTENTS. 



DIFFERENTIAL CALCULUS. 
PART I. 

SECTION I. PAOK. 
General Principles and Definitions 7 

SECTION 11. 
Differential Co-Efficients of Explicit Functions of a Single Variable, 23 

SECTION III. 
Differential Co-Efficients of Inverse Functions, Functions of Func- 
tions, AND Complex Functions of a Single Variable 41 

SECTION IV. 
Successive Differential Co-Efficients CO 

SECTION V. 
Relations between Real Functions of a Single Variable and their Dif- 
ferential Co-Efficients. — Taylor's and Maclaurin's Theorems . . 69 

SECTION VI. 
Expansion of Functions 90 

SECTION VII. 

Application of some of the Preceding Series to Trigonometrical and 
Logarithmic Expressions 99 

SECTION VIII. 

Differentiation of Explicit Functions of two or more Independent 
Variables, of Functions of Functions, and of Implicit Functions of 
Several Variables 117 

SECTION IX. 

Successive Differentiation of Functions of two or more Independent 
Variables, and of Implicit Functions i:v> 

SECTION X. 

Investigation of the True Value of Expressions which present Them- 
selves under Forms of Indetkrmination \:y2 

SECTION XL 

Dictrrmination ok thk Maxima and Minima Values of Functions ok one 
. Variable 17G 

SECTION XII. 
Expansion of Functions of two ok inkmje Independent Variables, and 
Investigation of the Maxima and Minima of such Functions. . . iss 

SECTION XIII. 

Change of Independent A'ariables in Differentiation JU 

SECTION XIV. 
Elimination of Constants and Aubitrary Functions by Dikki ki ntiavion, cjo 

5 



6 CONTENTS. 

PART II. 

GEOMETRICAL APPLICATIONS. 

SECTION L 
Tangents, Normals, -Sub- Tangents, and Sub-Normals to Plane Curves . 238 



249 



SECTION IL 

Asymptotes of Plane Curves. — Singular Points. — Concaat;ty and Con- 
vexity 

SECTION in. 
Polar Co-Ordinates. — Differential Co-Efficients of the Arcs and Areas 
of Plane Curves. — Of Solids and Surfaces of Revolution . . .266 

SECTION IV. 

Different Orders of Contact of Plane Curves. — Osculatory Curves.— 
osculatory circle. — radius of curvature. — evolutes and involutes, 281 



INTEGRAL CALCULUS. 

SECTION L 

Meaning of Integration. — Notation. — Definite and Indefinite Inte- 
grals. — Direct Integration of Explicit Functions of a Single Varia- 
ble. — Integration of a Sum. — Integration by Parts. — By Substitution, 315 

SECTION IL 

Integration of Rational Fractions by Decomposition into Partial 
Fractions 343 

SECTION in. 

Formula for the Integration of Binomial Differentials by Successive 
Reduction .360 

SECTION IV. 
Geometric Signification and Properties of Definite Integrals. — An- 
other De:monstration of Taylor's Theorem. — Definite Integrals in 
WHICH One of the Limits becomes Infinite. — Definite Integrals in 
which the Function under the Sign / becomes Infinite. — Definite 
Integrals that become Indeterminate.— Integration by Series . 874 

SECTION V. 
GEOMETRICAL APPLICATIONS. 
Quadrature of Plane Curves referred to Rectilinear Co-Ordinates. — 
Quadrature of Plane Curves referred to Polar Co-Ordinates . . 392 

SECTION VL 
Rectification of Plane Curves 405 

SECTION VIL 
Double Integration. — Triple Integration . . 411 

SECTION vm. 

Quadrature of Curved Surfaces. — Cubature of Solids 417 

SECTION IX. 

Differentiation and Integration under the Sign/.— Eulerian Inte- 
grals. — Determination of Definite Integrals by Differentiation, 
and by Integration under the Sign/ 429 

SECTION X. 
Elliptic Functions 465 



DIFFERENTIAL CALCULUS. 



-<=><><=>- 



FJ^ E. T IF- I E. S T. 



SECTION I. 

GENERAL PRINCIPLES AND DEFINITIONS. 

1, In the branch of mathematics of which it is now pro- 
posed to treat, we have to deal with two classes of quantities, 
— constants and variables : constants, which undergo no 
change of value in the investigations in which they are 
involved ; variables, which may pass through all values 
within limits that may be restricted or indefinite. 

Variables are usually represented by the final letters of the 
Roman alphabet; and constants, by the first letters of this, 
and sometimes, also, of the Greek alphabet. 

2, When variable quantities are so connected, that, one or 
more of them being given, the values of the others become 
fixed, the latter are said to be functions of the former, wliicli 
are called the independent variables, or simply the variables. 
The functions arc also called (Ic/)V)idc)tt variables. 

Thus, in the equation 

1/ = ax'^ -\- bx -f- c, 

y is a function of x, and in this case becomes not only fixed, 

7 



8 DIFFERENTIAL CALCULUS. 

but known, so soon as a value is assigned to x. So also, in 

the equation 

y = ax''- -\-hx -\- cz + c? , 

?/ is a function of the two variables x and 2, and is known in 
value when values are given to x and z. 

8, An J^xplicit Function is one in Avhich the depend- 
ent variable is given directly in terms of those which are re- 
garded as independent. 

In the examples given above, ?/ is an explicit function of a:; in 
the first, and of x and z in the second. In general reasoning, 
when we are not concerned with the particular form of the 
function, explicit functions are denoted by the symbols 

y = F(x), y =/{x), y = (p (x,z'), &g. 

4, An Implicit Function is one in which the relation 
between the function and the independent variable or variables 
is expressed by an equation that has not been resolved in 
respect to the function. 

Thus ax -\- by -\- c =z , 

ax^ 4- hxy + c?/l+ dx + ey +/= 0, 
x^ — az'^ -\~ cy^-xz ^ t/ = 0, 

are equations which require solution to render the variable, 
taken as dependent, an explicit function of the independent 
variables. Such functions are also designated by the symbols 

F(x, ?/) — 0, r/ (x, ?/, 2) — 0, etc. 

o. Functions are also classified, in reference to their com- 
position, into simjole or compound, according as they are the 
result of one or of several operations performed on the varia- 
bles. They are algebraic, when, in the construction of the 



GENERAL PRINCIPLES AND DEFINITIONS. 9 

function, the only operations to which the variables are sub- 
jected are those of addition, subtraction, multiplication, divis- 
ion, involution denoted by constant exponents, and evolution 
denoted by constant indices ; transcendental, when, in the 
composition of the function, the variables have been subjected 
to other operations, combined or not with those regarded as 
algebraic. 

Thus y ^=. a-^, y = log. x^ y z=. sin. x, y ^= sin.-^o;,'^ 

are examples of transcendental functions, and are exponential, 
logarithmic, or circular, depending on the mode in which the 
variable enters the functions. 

S. A function may be continuous or discontinuous. It is 
continuous, when, by causing the variable to pass gradually 
from any value to another separated from the first by a finite 
interval through all the intermediate states of value, the func- 
tion will itself pass gradually through all the values interme- 
diate to those corresponding to the extreme values of the vari- 
able ; and when, besides, the law of dependence of the function 
upon the variable does not change abruptly in the interval. 

y ~F{x) is continuous, if, by giving to x the infinitely small 
increment h = ax, y receives the infinitely small increment 
Ay — F{x -^ ax) — F{x). AVhen the law of the function is 
such that these conditions ai-e not satisfied, the function is 
discontinuous. 

7. The Limit of a Function is tlio value towarils 
which it converges, and from whicli it finally diU'ors by less 
than any assignable value, Avheu the variable upon wliie'h it 
depends itself converges towards some iixed value. 



* T\e:i(l arc whoso sine is .r, ami IVocjiuMitly writtou arc (sin. = x) . Tlic notations 
cos.-^r, tan.-'.r, ^^c, liuvo like sio-nitications. 



10 DIFFERENTIAL CALCULUS, 

It is of the highest importance that we should have a clear 
conception of the nature of limits as above defined, as this 
conception is at the foundation of the differential calculus as 
developed in the following pages. The following examples 
will illustrate the meaning of Umit^ and give distinct notions 
on the subject to those who have not already formed them : — 

1st, In the geometrical series 

J+i + J+Ac, 

the sum 8 of the first n terms is given by the formula 

-5- i_^ Y^ ^'■' ' 

and it is obvious, that, as n increases, (J)" decreases; and, when 
n becomes greater than any assignable quantity, (i-)" becomes 
less than any assignable quantity. In the language of the 
definition, as n converges towards infinity, S converges 
towards unity. Hence the limit of the sum of this series, 
when n is indefinitely increased, is 1. 

. 2d, The ratio of an arc of a circle to its sine has unity for 
its limit when the arc converges to zero ; that is, limit 

sm oc 

' — '- — :=1. For it is plain in the first place, that, for sensible 

values of x, the sine is less than the arc. And again : since the 
triangle formed by the radius, the tangent, and the secant, has 
for its measure J H. tan. x, while the corresponding sector is 
measured by J i?. x, it follows that the arc x is less than tan. x. 
Therefore 

sin. X X sin. x sin. x 

<" - r^ 1 ^ — cos. x. 

XX X tan. X 

sm X 
Hence we conclude, that, for sensible values of the arc x, 

X 



GENERAL PRINCIPLES AND DEFINITIONS. 11 

is always included between two ratios, both of which have 
unity for their limit. It must, then, have the same limit ; and 

we nave lira. = 1. 



Again 



sm. X 

cos. X 



sm. X cos. X tan. x 

X X X 

-. sin. X . ,. tan. x -, ■,• 

lim. =: 1 nr lim. COS. X : but lim. cos. xz^l, 

X X 

therefore the limit of — '- — must also be unity ; i.e., the limit 

X 

of the ratio of an arc to its tangent is unity. 

Cor. The limiting ratio of the arc to its sine, and of the arc 
to its tangent, each being unity, it follows, that, when the arc 
is infinitesimal, the arc and its sine, and the arc and its tan- 
gent, may be regarded as equal. 

X 

3d, For another example, let us take y =z , and trace 

out the series of values which 7/ assumes when positive vahies 
are given to x. Beginning with x := Q, we have y z=i 0. Bv 

division, the value of ?/ takes the form 1 — - ; from which it 

' ^ x+\ 

is seen, that, as x increases, the subtractivo part of the vahu> 
of y decreases, and y itself increases ; and, as x approaches 
-J- oo, ?/ approaches its limit 1 : but, for all finite positive vahies 
of a^, the values of y are less than 1. The difterence between 
y and this limit can be made as small as we please by giving; 
to X a value sufficiently great. Thus, if we wish to make this 

difference less than — - we make .r -. 1. 000.000. 

lu this same example, let us now give to x negative values, 
and observe the changes in tlu^ N-ahie of// as .r increases ne«;-a- 



12 DIFFERENTIAL CALCULUS. 

tivelj from 0, and approaches — oo . Replace x by — t, then 
X __ — / t 

^ ^ ^"+^1 ~~ — ^ + 1 ^ r^"T' 

and let us consider the values of y answering to valnes of t 
between the limits and 1. Beginning with ^ = 0, we have 
y z=zO: for all other values of t between these limits, the 
denominator of y being negative, y is itself negative. As t 
increases, y increases numerically ; and, when t differs from 
unity by less than any assignable quantity, y is greater 
numerically than any assignable quantity ; that is, — oo is 
the limit of y for ^ = 1, which answers to x = — 1. This is 
equivalent to saying that y has then no finite limit. 

When t passes 1, the denominator of the fraction be- 

t — i 

comes positive, and y changes from negative to positive. In 
this case, y passes abruptly from — oo to + °o while t is pass- 
ing through the value 1. The value of y may now be put 

under the form y = -. For all finite values of t greater 

than-[-l, y is greater than 1 : but y decreases as t increases; 
and finally, when t becomes greater than any assignable quan- 
tity, y will differ from its limit, unity, by less than any assign- 
able quantity. 

Trigonometry furnishes a case of limit similar to that of 
this example, when t passes through the value unity. As an 
arc increases continuously from 0, its tangent also increases 
continuously, but more rapidly than the arc ; and, as the arc 
approaches 90°, the tangent approaches its- indefinite limit 
-|- oo . When the arc passes through the value 90°, the 
tangent changes suddenly from an indefinitely great positive 
to an indefinitely great negative quantity. 



GENERAL PRINCIPLES AND DEFINITIONS. 13 

S, The exact meaning of the word " limit " will be under- 
stood from what precedes ; but it is well to call attention to 
abbreviations of expression frequently used in this connec- 

sm oc 
tion. In findins; the limit of — '- — when x is diminished with- 

X 

out limitj it would be said, 

. sin.- cc 
limit r= 1 when x z=zO: 

X 

but it must be borne in mind that — '- — cannot reach this 

X 

limit so long as x has any value. And, if we actually make 

. sin. X . . „ 

a? = 0, the ratio has no meaning ; in fact, ceases to exist. 

X 

It is true, that if x be not supposed to vanish, but simply to 
differ from by less than any assignable quantity, that is, if x 
becomes infinitesimal, the ratio retains its significance, and its 
value will differ from its limit unity by less than any assign- 
able quantity. 

In this case, the language is an abbreviation for this or its 

equivalent : ^' As ::c is diminished, the ratio converges 

X 

towards unity, and can be made to differ from it by as small 
a quantity as we please by taking x sufficiently near zero." 
And, in all similar cases, the language is to be interpreted in 
the same way. 

In other cases of limits, the inconsistency just pointed out 
does not present itself Any finite value of yy, in the example 

X 

11 =1 -, that answers to an assumed and finite value o( j\ 

^ X + V 

may be taken as a limit of i/ ; and it would bo strictly correct 

to say 

X 1 

limit of = - wlien .r — 1, 

X -\-\ 2 

This corresponds to llie delinition of limit given in Art. 7. 



^ , X oo , 0^ ± 1°^, will be established in a subsequent 



14 DIFFERENTIAL CALCULUS. 

0, Rules for the evaluation of functions, which, for particu- 
lar values of the variable, assume the indeterminate forms 



section : but it is necessary for our purposes to consider in this 

1 
place the function ?/ = (1 -|- ^Y, and to find its limiting 

value when x = 0; the function then taking the indeterminate 

form 1^. 

The variable x may converge towards its assigned limit 

zero through either positive or negative values. Let us first 

suppose X to be positive, and represent it by the fraction - ; 

m 

then, as x diminishes, m increases ; and, when x becomes a very 
small quantity, m becomes a very great quantity. 

If m be an entire positive number, we have, by the Binomial 
Formula, 

m m-1 m-2 1 

a development which will contain m -\- 1 terms. 

Dividing both numerator and denominator of each term by 
the power of m that enters the denominator, we find 



l + x). = (!+-) ^2+{(i--)+,8(i--:.){i-, 



111/ ]\/ 2\ /, 3\ 

Under the hypothesis that m is a positive whole number, the 

12 3. 

expressions 1 , 1 , 1 , &c., will each be positive, 

m m m 



GENERAL PRINCIPLES AND DEFINITIONS. 15 



and less than unity. Therefore {\-\- — ] = 2 -|- some posi- 
tive quantity ; that is 

Again : the development will be increased in value both by 

12 3 

nearlectins: the subtract! ve terms — ? — j —? <fec., and also by 

replacing each of the denominators 2, 3, 4, <fec., by the least 
denominator 2 ; that is, the true value of the development is 
less than 2 plus the series 

1 1 1 

2 ^ 4 ^ 8 ^ 
But this series cannot exceed 1, however far continued: 

therefore (l-j-^j-^^^flH ) i^ always included between 



the limits 2 and 3. 

If — ^=^m is a fractional number, it will be found between 

X 

two consecutive whole numbers m and 7^ z= ?m -|- 1. Let s and 
t be two positive proper fractions, whose sum is always equal 
to 1, and make 

1 ^< — 

-•- , ,1 m 

— = m 4- 5 = ?i — /, w4ience ^ j 

X ^ 1 

and the expression ( 1 + •^' )'" ^""^ ^^^ includod between 



IN. L . u ^ A .n l-^ 



m. / V 'm 



and 



' + «l*-l'+7j -|i' + . 



16 DIFFEREXTIAL CALCULUS. 

Now, as X decreases indefinitely, m and n increase to infin- 
ity, and the two quantities M + - j and M + - V both con- 
verge to the same limit, which, as was proved above, is includ- 

. s t 

ed between 2 and 3; while the exponents 1 -\ , 1 — — , 

conver2:e to the limit 1. 

. / 1\1 
It follows, therefore, that the two expressions ( 1 -| y^ 

1 \1 

1 _i- — \x have the same limit, and that this limit is the same 

/ 1\'" 

as that of ( 1 + — ) when m, regarded as a positive whole 

number, is indefinitely increased. 

Finally, if x is negative, and either entire or fractional, make 

X =^ — : so that, for all values of x numerically greater 

than 1, z must be negative and included between the limits and 
1 ; but, for values of x less than 1 (and it is with these alone 
that we are now concerned), z must be positive, and increase 
to infinity as x decreases to zero. Making this substitution 
for X, we have 

Hence, when x approaches its limit zero through negative 
values, the limit of the expression (l-f-^)"^ == limit 

is the same as when this limit is reached 
by causing x to decrease through positive values. 




GENERAL PRINCIPLES AND DEFINITIONS. 



17 



To find what this limit is, resume the equation 



i+.)i=2+^fi-iv^i(i-iyi-^ 



m 



111,, 
1111, 



m \ m 



mj 
1 - 



m J 



m 



m 



+ 



m \ m 



2345 

and suppose m to be infinite ; then 

.. A \1 1 11 111 1111 

hm. M 4-^]^- .^2 + ^+23 + ^^^+^^^^+^''-^ 



2 

1 

T 

1 
2T3 

1 

1 
2X4^ 

1 

2.3.4.5.6 

1 

2.3.4.5.6./ 

1 
2.3.4.5.6.7.! 

1 



2.3.4.5.6.7.8.0 



23 4 ' 2 3 45 
:= 2-0000000 . 

= -5000000 . 

= -0416666 . 

:= -0083333 . 

= -0013888 . 

= -0001984 . 

= -0000248 . 

3= -0000027 . 



Sum =r 2-7182818 . . . , a numlun- that is iiu 
with unity. 

It is iho base of the Napierian system of K 
will be denoted by c : 
3 



>mnuMisui-abh^ 
-;-;irithius, and 



18 DIFFERENTIAL CALCULUS. 

Therefore 

The symbol I will be hereafter used to designate Napierian 

logarithms, and L will denote other logarithms. 

1 
10, Taking the Napierian logarithm of (1 -\- x)^, we have 

2(1^ x)7- ^ 1 ? (1 4- x) = -^^1±5^ ; 

X X 

and, passing to the limit by making x 1= 0, we have 

l(\ A- x) i 

lim. ^ ^ ' = lim. 1(1 + j:)-r = le = 1. 
a:; 

In any other system of logarithms, we should have 

X X 

and at the limit 

L(l 4- x) 1 1 

lim. _i_J_l_{ — liin. L{l+x)^ = Le = j-, 



X 



la 



a being the base of the system characterized by L, and ob- 
serving that, since the logarithms of the same number in two 
systems are as the moduli of those systems, we have 

Le : /e :: M : 1, or Xe : 1 :: J/ : 1 
La\la :: 31 : 1, ov I : la:: 31 : 1\ 

and therefore Le z= 31 ^= — the modulus of the svstem of 

la 

which a is the base. 

i_ 

11, Since lim. (1 -}- x)^ = e when x is decreased without 

i 

* The notation (i 4. -jA ■'■ indicates that the value of the expression correspond- 
ing to .r = is taken. 



GENERAL PRINCIPLES AND DEFINITIONS. 19 

1 
limit, we can from this deduce the limit of (1 -|- az)'^ in whicli 

a is an}^ constant quantity. 
Thus, 



Xow, as z diminishes without limit, az Avill also diminish 
without limit ; and therefore 

lim. (1 -[- az)'"' =^ e ; 

lim. (l -\^ az)^ = e«. 
12. In any system of logarithms, 

z 

lim. — ^— i— ! — lim. Z( I + zY = Le ; 

z - I - 

and, if the logarithm be taken in the Napierian System, 

l™. Ki+l) = /« = !. 

z 

13, Resuming the equation 

2 

and making 1 -\- z =^ a^', whence (taking logaritlims in tlie sys- 
tem of wliich a is the base), r r= Z(l -h c) and c : - a'' — 1 ; 
therefore 



L{\ + ■-)'=' 



V 



a"— 1 



or, by taking the reciprocals, 



1 a' — 1 

1 



X(l +Zr: V 

Now, as c dimiiushes without limit, so also will /•, and they will 
reach the limit zero tog(4hor : (herc(i>ro 



20 DIFFERENTIAL CALCULUS. 




but lim. 



V 

Suppose a = e"', whence m=^la; 
and therefore 

g mv -j^ 

lim. = Ze'"=:m. 

V 

14, To define some of the terms, and explain the meaning 
of some of the symbols, employed in the calculus, let us take 
the explicit function of a single variable 

and give to x an increment denoted by ax ; y will receive the 
corresponding increment 

Ay= a/(^) =:/(^ + ax)— /(a?), 

and therefore 

A^ ^ f{^j:J^^x)—f{ x)^ 

AX AX 

When AX = 0, the ratio ■ — takes the form tt : yet it has in 

AX ^ * 

fact a determinate value, which is generally some other func- 
tion of cc, and expresses, as will be seen presently, the tangent 
of the angle that a straight line, tangent to the curve of which 
y =f[x) is the equation, makes with the axis of the variable 
X. This limiting value of the ratio of the increment of the 
variable to the corresponding increment of the function is 
called the differential co-efficient, or derivative of the func- 
tion, and is represented by the notations 

/ r// X ^^ r ^^^ V f{x^Ax) — f(x) 
y' fix), -r- , lim. — = lim. ^-^ ^ '- — ^i-i— \ 

-^ '-^ ^ ^' dx' AX AX 



GENERAL PRINCIPLES AND DEFINITIONS. 21 

It is to be observed that the characters a, c?, are not factors, 
but symbols of abbreviation ; the former signifying increment, 
difference, or change in value, without reference to amount : 
while the latter is restricted to particular increments called 
differentials^ having such values, that the ratio of the differ- 
ential of the variable to that of the function is equal to the 
differential co-efficient or derivative of the function. The 
differentials are usually regarded as infinitely small. 

1^, Before the ratio — reaches its limit fix), it must 

differ from it by some quantity which is a function of ^x, and 
which vanishes when Aa:^ = 0. We may therefore write, 

ACS- li ~J{-^) + ,, 

and, by clearing of fractions, 

^y =fi'^ + ^^) — /(•^) =f{x)Ax + yAX. 

.At/ 

From this it is seen, that, as the ratio — ^ approaches its limit 

/^(x), y must approach zero; and when ax, and consequently 
A//, becomes infinitely small, / must also become infinitely 
small, and should therefore be neglected in comparison witli 
the finite quantity /'(x). We shall then have 

It is therefore true, that, when the (hll'ereutial oi' a function 
is infinitely small, it is sensibly ecpial to the increment oi' {\\o 
function. 

These considerations are o( iniptu-tance, and ari^ nuulc bN 
many authors the basis of the deiinition of the diireriMitial 
of a function ; viz., ''The dilVerential of a. function o( a sin- 



22 DIFFERENTIAL CALCULUS. 

gle variable is the first term in the development of the dif- 
ference between the primitive state of the function and the 
new state which arises from giving to the variable an incre- 
ment called the differential of the variable ; the development 
being arranged according to the ascending powers of the in- 
crement.'' 

16, The definition of the difterential of a function follows 
from that of the differential co-efficient. It is the product of 
the differential of the independent variable by the differential 
co-efficient of the function. 

The object of the differential calculus is to explain the 
modes of passing from all known functions to their differential 
co-efficients, and the application of the properties of such co- 
efficients and the corresponding differentials to the investiga- 
tion of various questions in pure and applied mathematics. 

The operation of deriving from functions their differential 
co-efficients is called differentiation. 



SECTION II. 

DIBTERENTIAL CO-EFFICIENTS OF EXPLICIT FUNCTIONS OF A SINGLE 

VARIABLE. 

^7. It will be convenient^ before proceeding to establish 
rules for finding the differential co-efficients of the different 
kinds of explicit functions of a single variable, to investigate 
certain principles which are applicable to all forms. 

Constants connected with functions by the signs plus or 
minus disappear in the process of differentiation. 

The increments of the function and the variable will be char- 
acterized by the symbol A when they are written in the first 
members of equations ; but the labor of making the transforma- 
tions sometimes required in the second members Avill be les- 
sened by representing the increment of the variable by the 
single letter A, which will, of course, be equal to a^. 

Let y =/[x) i c, and give to the variable in this equation 
the increment A; then 

tlicrefore A y =/(x- -|- h) —J\x), 

Ay^ f{-r-^h)-f ix)^ 
A X h 



Passing to the limit by making A.r = A =r 0, 

\x dx ^ ^ ' 
and dy =z/\x) dx. 



24 DIFFERENTIAL CALCULUS. 

This differential co-efficient is manifestly the same as that 
which would have been found had there been no constant 
united iof(x) by the sign plus or minus. As, from their very 
nature, constants admit of no change of value, c has the same 
value in the new that it had in the primitive state of the func- 
tion, and must therefore disappear in the subtraction by which 
the increment of the function is obtained. 

dy 
Ex.1. yz=za-{-x, -- == 1, dy z^ dx. 

CLX 

dy 
Ex. 2. y z^ a — X, --_- =1 — I, dy ^z — dx. 

CLX 

18, If a function of a variable be multiplied or divided by 
a constant, the differential co-efficient will also be multiplied 
or divided by the constant. 

Let y=zcf{x), 

then y-^Ay = c/{x + h), 

Ay.:.cf(xJrJi)-c/{x) = c(f{x + h)-f{x)), 

AX h 

Passing to the limit, 



im. 



AX ' ^ ^ dx^ 



and 

Again : let 

then 





dy ~ cf {x) dx. 




y = \f{^), 


Ayz 


=:^^(/(^ + /0-/(^)), 


Ay__ 


l(/(:r-f /o-/(x-)); 


AX 


" c h 



AX dx c 



DIFFERENTIAL CO-EFFICIENTS. 25 

and dy =r ~f'{x) dx. 

dy 
Ex. 1. y z=. ax ^ h, -- =1 a, dy = adx. 

(XX 

^ r. X ^ dy 1 ^ 1 - 

Ex. 2. y = 0, ~ — -, dy =z- dx. 

a dx a a 

19* The differential co-eflScient of the algebraic sum of sev- 
eral functions of the same variable is the algebraic sum of the 
differential co-efficients of the separate functions. 
Let y =f{x) ±.(f{x) ±.^p{x) ^ . . . , 

then y J^ ^y—f{^x-\-1l)±:(p{x-\-h)±.'^J{x-^h)^ . . . 
Ay =/(x + h) ~f{x) :^(cp(x^ h) - q,{x)^ 

±:(^W(x + h)—xp{x)^^ . . . 

AX h h 

^ xpjx-l-h) -ipjx) 

h rtr . . . . 

whence, passing to the limits, and using the previous notation, 

and dy=f'{x)dxAz(^'{x)dx:^\v'{x)dx-±:_ . . . 

[f\x) ± q' {x) ± li^ (x) ± . . . ) dx. 



Ex. 1. 2/ = ax — hx -\- c 

V z=a — h, du z=z (a — b) dx. 
dx 7 ./ \ 

Ex. 2. y = a/{x) dz h^/^^ g [x) 



(/// = far {x) i b^/~^^ q' {x)) d.v. 



26 DIFFERENTIAL CALCULUS. 

20. The differential co-efficient of the product of two 
functions of the same variable is the sum of the products 
obtained by multiplying each function by the differential co- 
efficient of the other. 

Let y —f{x) Xq{x), 

then y -{- Ay =/{x -\- h) X cf{x -{- h) 

Ay =/(-^ + /O X>(x + h) -f{x) X qix) 
= (/(^ + h) -f{x)):p{x + h) + ((p(x + h) - cf{x))/{x)', 

AX ih 1 1 

Passing to the limit, by making ax = 7i i= 0, we see that 

lim. /(^±_^^lZ(^) zzr/^(x), lim. q:{x + h) =cp{x) 
li^,l(^^±A^^^lM ^^.(^): hence 

lim. ^=.^=zf{x) xq{x) + cp^(x) X /(x). 

A X (XX 

Dividing this equation by y ~f{x) X q\x), member by 

member, we have 

dy 

dx f'{x) cf' {x) 

21. The rule just demonstrated for finding the differential 
co-efficient of the product of two functions of the same varia- 
ble may be extended to the product of any number of 
functions. 

Let y =f{x) X q{x) X -V^(^), 

and make F[x) =cp{x) X V'(^) 5 

then y =f{x) X F{x), 

and ^^ =f (x) X F{x) J^ F^ (x) X f(x) ; 

but n' = F'{x) — (f/(x) X n^ix) -\-w'{x) X ^{x). 



DIFFERENTIAL CO-EFFICIENTS. 27 

Substituting, in the value of ^^ for F{x) and F\x), their 
values, we have 

This process has been carried far enough to discover the law, 
that the differential co-efficient of the product of any num- 
ber of functions of the same variable is the algebraic sum of 
the products found by multiplying the differential co-efficient 
of each function by the product of all the other functions. 

Ex. 1. y = (a -f- bx) (c — ax) mx 

,- z= b{c —ax) mx — a (a -|- hx) mx -|- m (a + bx) (c — ax) 
= m(ac -\- {2bc — 2a'^ — 3abx)xj 

di/ =^ m (ac + (2^c — 2a'^ — €>abx)x)dx. 

22, The differential co-efficient of the quotient of two 
functions of the same variable is equal to the divisor multi- 
plied by the differential co-efficient of the dividend minus the 
dividend multiplied by the differential co-efficient of the 
divisor, the result divided by the square of the divisor. 

^ ^p{x^h) cf{x) 

^f(x-\-h)cf(x)-cf(x + h)f( x). 
(fix + h) (jr(ir). 

z= (/(.r + h) _/(.r))(-K.r)-((r(x- + h) - q^x)) ;\x) ^ 



therefore 



AX h ^' h ^* 

(f{x-\-h)qi^x) 



28 DIFFERENTIAL CALCULUS. 



Passing to the limit, by making a^c = A z= 0, 
This result may also be obtained thus : 





y=f$y ■'• /w=2/<^(^); 


therefore, 


by Art. 20, 




f'{x)^y'^{x)+^>'(x)y■, 


therefore 






= /'(x),^(x)-f(x)<f'{x) 




(?(-)y 


Ex. 1. 


a-\-bx 
y -hJ^ax 




dy _{b + ax) & — (a + bx)a _ h^ — 0} 



dx (^ + ^^y (^ + ^^Y 

dy = — — ^ ax. 

23, The rules which have been thus far demonstrated in 
this section are independent of the form of the functions char- 
acterized by the symbols/, q), ip, &c.; and it has been assumed 
that the differential co-efficients of these component functions 
of the compound functions considered can be obtained in 
all cases. Before showing that this assumption is correct, by 
the actual differentiation of all known forms of simple functions, 

it is proper to make a few observations on the symbol -^ , used 

ax 

to denote the differential co-efficient of the function y of the 
variable x. 



DIFFERENTIAL CO-EFFICIENTS. 29 

111 the doctrine of limits, --^ represents the limit of tlie ratio 
— ^ ; and it must be borne in mind, that, at this limit, a x, and 

consequently a ?/, become zero. There would be, therefore, an 
inconsistency in viewing dx and dy as the representatives of the 
terms of a ratio that have vanished, until it be proved that the 
ratio itself does not also vanish. If the ratio remains, although 
its terms disappear, then dx and dy may be taken as indeter- 
minates, having for their ratio the final ratio of the vanishing 
quantities. This is the view to be taken of the differentials dx 
and dy^ according to definition. Art. 14, and which justifies us 

in regarding these differentials as the terms of a fraction in -,- • 

(XiX 

24:, Analytical geometry furnishes instructive illustrations 
of the meaning of differential co-efficients, as was intimated in 
Art. 14, and suggests many useful applications that can be 
made of the doctrine of limits. 

Whatever may be the nature of the function y i=/(x), every 
value of X that will give a real value for y will be the abscissa 
of a point of a curve of which y is the corresponding ordi- 
nate ; and, if the assumed value of x gives several real values 
for y, X will be the abscissa of a like number of points of the 
curve, having for their ordi- 
nates the several values of 
y. The curve is, therefore, 
the geometrical representa- 
tive of the relation between 
X and // in tlio equation y 

In the figure, suppose T 

SP'P to be the curve represented by the eqiuitiou // :=t\x\. 



y 




V 


/y\ 


T 

a 


/ 




A 






o 


^7 


ii 


I\ 


1' 


.\ 



30 DIFFERENTIAL CALCULUS. 

and let PM be a value of y corresponding to the assumed 
value OM for x; then give to x the increment MM'^^li, y 
will receive the increment F'Q ■=. ^y^ and we have 

F'Q^^y^f{^x+h)-f^x)^P'M'~PM: 
A ^ ^ /(: r + A)-/(^ ) ^ P;0^ 

!^expresses the trigonometrical tangent of the angle P'PQ^ 

which is the tangent of the angle that the secant line or chord 
PP' makes with the axis of the variable x. Now, it is evident, 
that, as liz=.MM' diminishes, the point P' moves along the curve 
towards P, and the secant line approaches coincidence with 
the tangent line TT' ; and finally^ when h vanishes, the coinci- 
dence of the 23oints,and of the secant with the tangent line is 
complete. The tangent line to the curve at the point P is then 
the limiting position of all secant lines which have P for one 

cli/ 
of the points in which they cut the curve, and , is the limit- 
ing value of the tangents of the angles that such secant lines 
make with the axis of x. 

25, The fraction — ^ always represents the ratio of the 

assumed change in the value of the variable to the corre- 
sponding change in the value of the function. These changes, 
when small, are properly called increments ; and it is evident 
that their ratio is the measure of the rate of the increase of 
the function to that of the variable : but it will be seen, that, 
for functions in general, this rate of increase will vary both 
with the initial value of the variable and the value of its 
increment ^x. If, therefore, the value of the increment 



^x 
equally so. But the conception of the limiting value of the 



were left arbitrary, the value of the fraction -^ would be 

^x 



DIFFERENTIAL CO-EFFICIENTS, 31 

ratio removes all "uncertainty, and suggests to tlic mind wliat 
the rate of change in the value of the function is, in the imme- 
diate vicinity of its value for any assumed value of the varia- 
ble. 

Ill the case of the curve, in the last article, the limit -, 

ax 

does not depend upon the increment ^x^ nor upon the form 
of the curve at finite distances from the point whose co-ordi- 
nates are {x,y), but depends only upon its shape and direction 
within insensible distancv.0 fiom that point. 

26. Let us apply these remarks to the equation 7/ =\/2px 
Avhich is the equation of a parabola referred to its axis, and the 
tangent line through the vertex as the co-ordinate axes. Giv- 
ing to X an increment, 

y + Ay — \/2p{x-f7i) 

Ay = y/2p (s/x -^h — ^x \ 



= V2i> — 



x^-\-li — X \ _ A\/2jo 



therefore lim. 



s/x-Yh^ ^x) y/x-\-li-^ ^x 

Ay S/^P 

^'^~^X^ll^^X 

All ^ ^y ^ V2/' V p_ 

A X dx 'i^x ~~ \/'lpx ~ y 



From analytical o-comctrv, wo know that — is the natui-al 

tangent of tlic angle that a tangent line to the parabola, at the 
point whose ordinate is y, makes with the axis of the curve. 

27* The dillereiitial co-eflicient of a I'uiu'tion, which is a 
power of the variable deuotcnl by any constant exponent, is 
the exponent multiplied by the variable with its original 
exponent, less one. 



32 DIFFERENTIAL CALCULUS. 

Let y — ^", 

then y J^ ^y — {x^llY 



^^ h 



X 



Put also {l + ty-l = z .-. (l+^)-z=l4-s. 



^y 

g inese suusliluliuhs ju lue uxpressiuii lur 
becomes 



Making these substitutions in the expression for -^- , it 



AX t 

Both t and 2 diminish with A, and reach the limit zero 
simultaneously with it. Taking the Napierian logarithms of 
both members of the equation (1 -|- ^)" = 1 -[- ^7 we have 

n.l{l-\-t) = l{l+z) 

n '= — - 

/(i + O* 

But, Art. 10, —-^ and -i_lLJ both have unity for 
2 t 

their limit ; hence 

But, since n is a constant, 

lim. -- = - lim. -; 

nt n t 

Is 2 

-lim. - = 1 .-. lim. 7=^-' 
n t f 



DIFFERENTIAL CO-EFFICIENTS, 33 

therefore lim. ^ =. lim. x'^-^ \ = ^f^ nx^-\ 

Lx i ax 

28. The rule of the last article ma/ also be demonstrated 
as folloAVS : — 

Let, as before, y z=lx'^^ 
then y -{- ^y ^=.{x -\-liY 

'x + li\ 



^x h h 

Now, whether n be a whole number or a fraction, positive 

or negative, it may be represented by , in which 2^; (7; and 

s are positive Avliole numbers. 

x -^h , 

Make = 2 .' , li = x{z — l)^ 

X 

and -^=x'"-i — r,-; 
therefore lim. - — =. lim. x"'^ ~ . 

AX- 2 — 1 

As A converges towards its limit zero, z converges towards 

the limit unity, and h and z reach their respective limits simul- 

r-q 

2" — 1 2-' — I 

taneously: Ave have then to find tlie limit of z=z " , . 

^ z — \z-\ 

Make u = z'; whence 2 ' = u^''^, 2 = ?t',and the limit of u 
is unity also, flaking these substitutions, we have 

2 * — 1 __ uP"l — 1 uP — ul uP — 1 — ( ii'^— 1) 

2 — 1 ~~ u'~\ ~ 'it\a' — \)~ n,\a'—\) 
Dividing both numerator and denominator (^'i this last trac- 
tion by u ~ 1, it becomes 

t^/'-^ - I- jt?"'^ -^ . , ^ -)- 1 — (/. ^'/-^ -f u'/-'^ -f . . . + 1 ). 
1^^ ("iT^i'+'it^ -f . . . + 1 )" * 

5 



34 DIFFERENTIAL CALCULUS. 



and; at the limit where w = 1, this reduces to / 

s 

T A2/ T 2" — 1 dy p — q ^ . 

hence hm. - - = hm. x"'^ == 7- = £c""^ — nx""'. 

i^x z — 1 ax s 

29. The rule proved in Arts. 27 and 28 is general ; bnt, 
when the exponent n is a positive whole number, the demon- 
stration below is more simple than either of those given. 

Let y =z x^ X X2 X • . . a;„, in which cCj,^.,, <fcc., are all func- 
tions of the variable x; then, Art. 21, 

r = f = 

^ dx 
^iXa^aX . . .a:„_i Xx\-^x^Xx,X . . . £C„_2 X x^X x'^_^-\- ko,. 
to n terms. 

iSow suppose x^=. X =. x.,_^=. . . . ^=: x.„, then x\^^ y-= 1 = 

x\_Y =^ • • . =■ x'] and each term in the value of y' becomes 

ic"'^: hence y^ = ^ ^=.nx'^~^. 
dx 

Under this supposition in reference to 7i, we may develop 

[x -\- li) ^ by the Binomial Formula, and thus get an expression 

for the ratio - — , of which the limit can be readilv obtained. 

^x 

Thus y = x'', i^y — {x-\-liY — x"" 

72-1 

= nx "-1 7i + 71 — ^— ic "-- /i 2 -[- <fcc. 
^^ r= 7? j;"-i -}- n ^^ — x'^'^h + &c., 

AX ^ 

in which all the terms in the second member after the first 

term contain li as a factor ; 

, ,. Ay dy „, 

hence lim. — ^ =r _^ = nx'^~\ 

Ax dx 



DIFFERENTIAL CO-EFFICIENTS. 35 

Ex. 1. y = « + hx\ -| = Zbx\ 

-CiX. ^. 7/ — ^„ = a;r-2. -^ = _ 



y -z=z —^z=L ax 



x^" ^ dx 



X 



3 * 



Ex. 3. 



dy 

2x 2aj3 2a 2^ 



(T 



■' + «' i^'^a^'' ~~ {X-' + a'^)"^ * 
SO, The differential co-efficient of a function of the form 

y^a^.a being a constant, is the function multiplied by the 

Napierian logarithm of the constant. 
^^^ y = a"", then 

and ^z.^a-^'ArJ 



Passing to limit, 

^^ = lim. . .^ ^^ll^^. .,..-!.- 

h 



Ay d?/ ^, a''~l n^ 1 



But, by Art. 13, 

hm. — ^ — — la : 

therefore ]\m.^'' = '^" ^a-rla 

A X dx 
^^' y = a''-%thcny=:(a^'y 

and 'f = a''-Ia\ 

dx 



Ex. 1. v=-^'^"+'-''^ ^^//__ (^+...,.0^^(^ + c-.r^)^ ^^ (,,,.,..) 
'dx~' dx ~ =-'-'' 



* The loftcM- ./ thus written before an expression i.ulic-ntos di.Vorontiation. Th,. 
,f T, = « -f /,,. J^ ,1^.,, _ / j^ eqnivjilont to - 



DIFFERENTIAL CALCULUS. 



^ n dy ri d. x^ , " 

Ex.2. v = ^^,-# = ^^ -^r- =^^^ c^ 
^ dx dx 

Ex. 3. ?/ == e^-^~i -f c-^^^i 



V-l (e-^-^-1-e-^ 



-^-1 



p]x. 4. y = e'^ = €'€■'. 

(XX 

31, The differential co-efficient of a function which is the 
logarithm of the variable taken in any system is the modulus 
of the system, divided by the variable. 

Let y^^Lx; 

X -\-h 
then y -]- Ay — L{x -]-7i), Ay —L{x-\-h) — Lx=:L-^—. 

X 

-^ x-\-h 
AVhence 



Ay _ 


L 


x-^h 

X 


AX 




h 



Make h = xz : therefore 

A^ _ Z(l + ^) _ 1 XjM- ^) 

AX XZ X Z ' 

But h and z reach the limit zero simultaneously ; and, by Art. 
10, the limit of ~T^J , for 2 = 0, is equal to Le =1 ^^ 

= 31, the modulus of the system : therefore 

, . A ?/ dii 31 

AX dx X 



Hence, 


if 


^ ^' dx~ X 


Ex. 1. 




1 dy 1 
y =:lx" =^ nix, -f^ =: n -. 
dx X 


Ex. 2. 




y = xHx, d^ — ^ -^2xlx. 



DIFFERENTIAL CO-EFFICIENTS. 37 

32, The differential co-efficient of the sine of an arc is 
equal to the cosine of the arc. 
Let y = sin. x, then 

y-\-^y = ^m. {x-\-h) 

t^y — sin. {x -f- h) — sin. x 

== 2 cos, ix -[- ~\ sin. I . (Eq. 16, Plane Trig.) 

. h 
sm. - 

2 

But, when A is diminished indefinitely, the limit of 
. h 



sm. ^ 










=: 1 


(Art. 7 


■), an 


id lim. c 


h 


2 










therefore 




lim. 


' AX ' 


dx~ 



COS. x. 

33. The differential co-efficient of the cosine of an arc is 
equal to minus the sine of the arc. 
Let y = cos. x, then 

y -h ^ // = ('O^. {x-\-h) 

Ay — COS. [^x -\-li) — cos. X 



Ay 

AX 



. // 
Sill. - 

At the Hiiiit " :^ 1, sin. ( .r + :| ) = sin. x 



-: sm. ,- sm. 

. h 
sm. - 

!<in 




A 


o 





38 DIFFERENTIAL CALCULUS. 

therefore 

,. Ml dii 
lim. — ^ = / =: — sin. X. 
^x ax 

34:, The differential co-efficient of the tangent of an arc is 

equal to 1 divided by the square of the cosine of the arc. 

Let y = tan. x, then 

?/ 4- Ay = tan. (x+A) 

A y = tan. (;x -\-h) — tan. x 

sin. {x -\- h) sin. a? 

COS. (x -{- h) COS. X 

sin. [x -f- //,) COS. X — COS. {x -\- h) sin. x 

COS. [x -\- h) COS. X 

sin. [x -\-h — x) 



therefore 



COS. {x -|- h) COS. x 

sin. h 
COS. {x -\- h) COS. X 

A ?/ sin. A 1 



(Eq. 8 Plane Trig.) 



AX Jl COS. {x -\-h) cos.x 



, ,, ,. ., sin. h . 1 1 
at the hmit — — zz= 1 =^ ; 

A. COS. {x -\- h) COS. X cos.^ a; 

hence lim. — "^ = ^ = ■ — z= sec.- x. 

AX ax COS.- X 

35, The differential co-efficient of the cotangent of an arc is 
equal to minus 1 divided by the square of the sine of the arc. 
Let y = cot. x, then • 

yArAy— cot. (x + A). 

Proceeding with this as in the case of the tangent, we should 

find 

dy \ 

-7- = .— -^r- = — cosec.'^a:. 

ax SYW.'- X 



DIFFERENTIAL CO-EFFICIENTS. 39 

36. The differential co-efficient of the secant of an arc is 
<)qual to the sine of the arc divided by the square of its cosine. 

Let y = sec. x, then 
y -\- ^y =z sec. (x -|- h) 

hy z= sec. {x + h) — sec. x 

^1 1 cos. x — cos. {x -\- h) 

COS. {x -j- h) cos. x COS. X cos. {x -f- h) 

2 Sin.- sin. a?+ - 

:=z -) "^ (Eq. 18 Plane Trig.). 

cos. X cos. {x -}- A) 

Therefore 

. h . h 

sin. - sin. a:; -|- - 

A ^ f 



A a; ^i COS. a:: COS. {x -\- h) 

2 

Passing to the limit, 

,. A 2/ dy sin. a? 

lim. —^ = -^ = ,^- = tan. cc sec. x. 

AX ax COS.'' ic 

37. The differential co-efficient of the cosecant of an arc is 
equal to minus the cosine of the arc divided by the square 
of its sine. 

Let y = cosec. x, then y -\- Ay =: cosec. (x -f- h), and A // =i 
cosec. (x -f- h) — cosec. x, and so on ; the process being the 
same as in the case of the secant. We should thus find 
dy cos. X . 

-j^ =: — . , =: — cosec. X COt. X. 

dx sin.-.r 

38. The dillorontial co-efficiont of the versed-sine of an arc 
is equal to the sine of the arc. 

Let y .:= vers, x :^ I — cos. x, then we have 

dy _ d. vers, x _d. (l — cos. x) _ d. cos. .?■_,• 
dx dx dx dx 



40 DIFFERENTIAL CALCULUS. 

39, The circular functions whose differential co-efficients 
have been thus far found are called direct circular functions. 
Since the tangent, cotangent, secant, and cosecant may all 
be expressed under a fractional form in terms of the sine and 
cosine, their differential co-efficients could have been found 
by the rule of Art. 22. Thu^ 



LS . 



sm. X 

1st, y = tan. x =: 

cos.a? 

dy _ cos.^ X -\- sin.'- x ^ 1 
dx COS.- X cos.'^ X 



COS. a? 

2d, y = cot. X = 

' ^ sm. X 



dy sin.- a; -f- cos.^ cc 1 

dx ~ sin.^ X sin.'^ x' 

3d, ,=.sec.. = -l- 

COS. X 

dy sin. x 

~ = r- =: tan. X sec. x, 

dx COS.- a; 

4th, y = cosec. x 



sm. a; 



C?Z/ COS. x 

- J z=i . — - — = cot. X cosec. X. 

ax sm.'' X 



The other forms frequently given to the differential co-effi- 
cients of the direct circular functions will be readily recognized 
by the student familiar with the elementary principles of trig- 
onometry. 



SECTION III. 

DIFFERENTIAL CO-EFFICIENTS OF INVERSE FUNCTIONS, FUNCTIONS 
OP FUNCTIONS, AND COMPLEX FUNCTIONS OF A SINGLE VARI- 
ABLE. 

4:0* The inverse circular functions are those in which the 
sine, cosine, tangent, &c=, are taken as the independent varia- 
bles, the arcs being the functions. Thej are written y =: 
sin.""^ x,y ^=z cos."^ ^7 ^ = tan.~^ x, &c.; and are read y equal to 
the arc of which x is the sine, cosine, tangent, &c. These 
functions are sometimes written y = arc sin. x, y = arc tan. x, 
<fec., and also y =: arc (sin. z:^ ic), ?/ = arc (tan. = x), (fee. ; but 
the first notation, being the shortest, and that generally adopted, 
will be uniformly used in what follows. 

4:1, If we have ?/ = (y(ic), then the differential co-efficient 
of cr, regarded as a function of?/, is the reciprocal of the differ- 
ential co-efficient of?/ regarded as a function of a:. 

That is, if y = i:p[x), then x must be some function of //, 

such as X — . u.(//) ; whence -^^ — if' {x), [^"^ = u-^ (^y) : and, ac- 

cording to the principle enunciated, we should have 

dx ,. . I 

As an example, take tlie equation 



from which we <ret 



t) 



2.2,.+ n. 



II 



42 DIFFERENTIAL CALCULUS. 

Solving the equation with respect to x, we have 



X = - 1 ± V2/ + 4 ; 

(^X 1 

whence -.,- =z i — ; but d= v^ 9 / 4_ 4 — ^ 4_ 1 • 

■ T ^ c?.:c 1 

therefore -^- = — — --—, 

which accords with the theorem ; and we will now prove that 
what holds in this particular case is true for all cases. 

Let y=zqj(x) . . . (1) be the given function; and since, 
from the nature of equations, x must also be some function of 
f/, suppose x=:w{i/) . . . (2) to be that function. 

If, in Eq. 1, x receives the increment ax, y will receive a 
corresponding increment a y : therefore 

2/ + Ay=:(^(x + A^) . . . (3). 

Now, Eqs. 1 and 2 are but different forms of the expression 
of a certain relation between the variables x and y ; and what- 
ever values of y in Eq. 1 result from an assigned value to 
X, if one of these values of y be assigned to y in Eq. 2, then, 
among the different resulting values of x, one at least must be 
the value assigned to x in Eq. 1. 

It is therefore proper to assume that x and y have the 
same values in Eq. 2 that they have respectively in Eq. 1. 
Change, then, in Eq. 2, y into y -\- Ay, and x into x -\- ax, 
these symbols having the same values that they have in Eq. 
3 : hence 

x-\-Ax = w{y J^Ay) . . . (4). 

From Eqs. 1 and 3, we have 

AJ__ Cf{x-{-Ax} — Cp{x) ^ ^ ,^.. 
AX AX 



DIFFERENTIAL CO-EFFICIENTS. 43 



from Eqs. 2 and 4, 

^^^ ^{y-{-^y)-v^{y ) ^ . . (6): 

Ly Ly 

multiplying Eqs. 5 and 6^ member by member, then 

Aa:; A?/ Acc hy 

By the preceding remarks, — - x — = 1 5 ^i^^; ^^^ ^^^^ 

AX A?/ 



^ = o)' (x) ; and the second factor, - :== li^'(x) : hence 
ax ay 



second member of Eq. 7, the first factor at the limit becomes 
id the i 
dy dx X w X 

dx 1 ,, , 1 

Whence "7- =-7/"~T = ^ (^) = ■;7~* 
dy cp'{x) ^ 

dx 
42, If we have 

y = xv{z) ... (1) 

z = ^ix) ... (2): 
tlien y is Qj function of x ; for, by substituting in the first of 
these equations the expression for z in the second, y becomes 
an explicit function of x. Suppose this to be denoted by 
y=fix) ... (3): 

Now, if X, in Fai. 2, receive the increment A x, z will take 
the increment az; and, in consequence of this increment oi^ z, 
y in Eq. 1 will become y -\- s y : hence wo should have, iVoni 
Eqs. 1 and 2, 

2/ + -^!/= Vi.^' + A:;) ... ^4) 
z J^Az — ir{.v-{-ix) . . . ^5); . 
also, from Ecj. 3, 



44 DIFFERENTIAL CALCULUS. 

From the mode of dependence of the variables, we may 
assume that the symbols x^ y, z, a x, A y, a z, have respec- 
tively the same values in all of the preceding equations in 
which they occur. Subtracting Eq. 3 from Eq. Q, member 
from member, and dividing both members of the result by ax, 
we have 

A f{x^Ax)—f[x) 



AX AX 

similarly, from Eqs. 1 and 4, 

Ay \p{z -\- Az) — U'(z] 



AZ AZ 

and, from Eqs. 2 and 5, 

AZ Cp (X -\- Ax) — (J) (x) 



AX AX 



■ (T); 



(8); 



(9). 



Multiplying Eqs. 8 and 9, member by member, we have, 
because the symbols are supposed to have the same values 
throughout, 

M/ ^ __Ay __ w{z-^Az) — u^jz) q.{x^Ax) — q (x) ^ ,^^y 

A Z AX AX AZ AX 

equating the second members of Eqs. 7 and 10, 
f{ x^Ax)—f {x) ^ w{z-^Az) — w{z) ^ q<{x-}-A x ) — q{x) , 

AX AZ AX 

Whence, passing to the limit, 

f{x) = w^{z)q^{x), 

dy dy dz 

^ dx dz dx 

Ex. 1. ?/ z= 2^ + 3s - 5 . . . (1), s = 2:^ — 3 ... (2) 

"^^-=2. + 3 "^- = 2 '^=.'^^ = 4, +6 
dz dx ' dx dz dx 

z=8a; — 6. 



DIFFERENTIAL CO-EFFICIENTS. 45 

By placing for z, in Ef[. 1, its value fronn Eq. 2, W(3 find 

dy 

y r= 4x^ — 6^ — 5, whence ,- — 8x — 6 ; 

the same result as was found by the first process. 

4z3. Differential co-efficients of the inverse circular func- 
tions. 

1st, Differential co-efficient oi y -— sin.^^cc. 
Since y = sin.~^ x^ x ^= sin. y ; and therefore, by Art. 41, 
dx 
dy 
and therefoi-e, by Art. 41, 



cos.^ = d= Vi — x'-^; 



dy 1 ^ 1 



c/x COS.?/ Vl — a;^ 

2d, Differential co-efficient of y = cos.~^ x. 

Here y — cos.-^ x gives x — cos.y: therefore. Art. 33, 

dx . / 

- = - sin. yz=z^^i_^2'^ 

dy 



and therefore, by Art. 41, 

dy _ 1 



= q= 



dx sin.y \/l — x'' 

It would bo superfluous to point out the necessity for the 
signs i; q:^, bcforo the dillerential co-efficients in this and 
the preceding case. 

3d, Differential co-efficient of // = tan.~^ .r. 

From y = tan.~^ x, we have .r := tan. // ; therelbre 

dx 1 

-.- = =ir sec.-' // z= 1 -}- tan.- // (Art. i> i) : 

dy cos.-?/ '■ 1 ^ V 

1 c??/ „ 1 

and - '- = COS.- y = , , - — — (Art. 41). 

dx "^ I -f- tan.-// ^ ^ 

Whence ; ::r: _ — ,. 

dx 1 -\-x' 



4 6 DIFFERENTIAL CALCUL US. 



-'x. 



4th, Differential co-efficient of y z=z cot.' 

From y = cot.~^ x, we have x = cot. y : therefore 

dx 1 - 

dy '^~li^'^ ~ cosec.^y = - (1 + cot'y) (Art. 35). 

^^=z-sin.2?/z=---,l-^ (Art. 41). 
dx ^ 1+x^ ^ ^ 

5th, Differential co-efficient of y := sec.~^ x. 

From y =: sec.~^ x, we have a:: := sec. y : therefore 

dx sin. 2/ 2 • / A ^ o^N 

and ^^ ^ c_os^ ^ - ,1 . (Art. 41 ). 

dx sm. y sec- y sm. ?/ 

But sec. y = , hence cos. y = = - ; and 

COS. y sec. 2/ a:^' 

1 . S^X' 1 

1 — sin.2 y __ __^ g|j^^ ^ = i ■ therefore 

iC X 

<^^ a: ViC-^ — 1 

6th, Differential co-efficient of y = cosec.~^a:. 

We shall merely indicate the steps. 

dx COS. y . , . «-x 

X = cosec. ?/, -7- = -. — r^ = — cosec. -y cot. y (Art. 3/ }: 

dy smJy i^ i/ v 

(^y 1.11 

-7-== , sm. ?/ = = - : 

ax cosec. y cot. y cosec. y x 

.-. Coty^rtVo^'^-l; .-. ;;^=^^ 



c?^ a; Va;2 - 1" 

7th, Differential co-efficient of ?/ = vers.""^ ic. 
Taking x for the function, we have 

X ^=z vers. 2/ = 1- — cos. ?/ ; 



DIFFERENTIAL CO-EFFICIENTS. 47 

doc dtt 1 
therefore -^ = sin. ?/ (Art. 38), and "^ = -, (Art. 41): 



but — L-=±— ^ z=± ^ 



sin.?/ Vl — C0S.2?/ Vl — (1 — vers.^) 



Vl — (T - x)2 V2x — x^ 

44. The principle demonstrated in Art. 41 has greatly 
simplified the investigation of the formulas expressing the 
differential co-efficients of the inverse trigonometrical func- 
tions. They may, however, be determined directly, without 
the aid of this principle. 

We will illustrate the manner in which this may be done 
by a single example : — 

Let y =r sin.~^ x, then 

y^Ay = ^mr\x-\-li)-^ 
and A^/ = sin.~^ (x -|- /i) — sin.~^x. 

The second member of this last equation is the difference 
of two arcs whose respective sines are x -\- h andx; and this 
difference is, by trigonometry (Plane Trig., Eq. 8), equal to 
an arc having for its sine the sine of the first arc multiplied 
by the cosine of the second, minus the cosine of the first 
multiplied by the sine of the second. Expressing \.\\o cosines 
of these arcs in terms of their sines, we have 
A?/ = sin.""' [x -f- li) — sin.~^ x 



Ay __ siii.-^ {i^x^h)^/T—{- - .r V' [1 - {X 4- hV-i) 



^x h 



48 DIFFERENTIAL CALCULUS. 

Make z — [x^ li) »^ 1 —~x' — x j^J \y — (p^ ^- ^0^] • 

therefore -^ = — =. = y 

^x li z 11 

Now z and h diminish together, and become zero simultane- 
ously. At the limit, — *- — =1. To find what - becomes at 

Z li 

the same time, multiply and divide the expression for z by 

[x -\- li)\/\ — x'^ -\- x^ (l — {x -\- h)')', then 

(^ + ky (1 _ ^2) _ ^2 A _ (^ _|_ 7^)2^ 

^* A^x+/^)Vl~-^^ + :^V'[l - (-^ + ^0']) 

(j: +/0\/n^~^^' + X v/(i - (^ + /O 

Pass to the limit by making h = 0, and we have 

,. z X 1 

lim. -^ = 



h Xa^I^x' \/\ — x'' 
^? — J^ 

4^. Differential co-efficients of functions of the form y = f' 
in which t and 5 are both functions of the same variable x. 

Taking the Napierian logarithms of both members of the 
equation ?/ = ^% we have ly ^ sit. By Art. 42, the differ- 
ential co-efficient with respect to x of ly is 

d^y ^\^y • and, by Arts. 20, 42, that of sit is 
dx y dx "^ 

^ d.s ds d.lt dt __ J ds 1 dt 

ds dx dt dx dx t dx 

Now, since the equation ly = sit is true for all values of x. 



DIFFERENTIAL CO-EFFICIENTS. 49 

the differential co-efficients of its two members must also be 
equal : therefore 

1 dy __jdssdt 

y dx dx t dx^ 

, dy I ^ds , s dt 

^ ^^"^""^ dx^^yy^dx-^tdx 

ax ax \^ ax t ax 

46, From an examination of the particular cases treated in 
Arts. 19, 20, 45, we deduce this general rule for finding the 
differential co-efficient of any compound function : Differen- 
tiate each component function in succession, treating the others 
as constant, and take the algebraic sum of the results. 

Kuies have now been given for the differentiation of all 
known forms of algebraic, logarithmic, exponential, and circular 
functions of a single variable ; and we have seen, that, in gener- 
al, the differential co-efficients of these functions are themselves 
functions of the same variable. 

47' The following exercises are given that the preceding 
rules may be impressed, and that the students may become 
expert in their application, and familiar with the forms of the 
differential co-efficients of simple and complex functions : — 



1. 


y = ax', ^^ = ^>«^'' (^Vrt. 28). 


2. 


y = abx^ ■ — ex'' 




^^=:^ahx'~2cx (Arts. 10, 28). 


3. 






dy _2ax{h — x'f + 6ax^ (h _ x')» 



^ = ^ fl^ZI-^y-^ '- (Arts. 2 

_ 2a.f(?) + 2.f-') 
7 ~~{b-x^y 



50 DIFFERENTIAL CALCULUS. 

4. y = VS+V^ip= (^ + (^ + c)^y 

Put z = X -\- {x -\- c)^ ] then y — z^, 
and ^ = ^^ (Art. 42). 



But 



dx dz dx 
dy__ J_ __ 1 

^'~2i^~2(^x+(x + c)iy> 

and | = l + l(. + er^^^(^+^^: 

therefore ^~ ^ , 3(a. + c)t + l 

^^ 2(x + (^ + cf)' 3(x+c)t 

__ 3 V(^+"cy+ 1 



5. 2^ = Z(a:; + \/l +/c"^). Make 2 = cc -I- Vl+cc^; 
then y = Iz, and -^- — -^-^ 



dx dz dx 



dy __\ 



But -!-=■-=■ 7== (Art. 31). 

dz z x-^-^X^x"- ^ ^ 

and ^^ = 1 [ ^ ^ a^ + VlT^" l 

therefore ^ = l__ ^+^H^ ^ ^ . 

dx x^^ij^x'' Vl+cc^- \/l+aj2 

The utihty of substituting a single symbol to represent a 
complicated expression before differentiation is exemplified in 
this and the preceding examples. Oftentimes the labor, both 
mental and mechanical, of the mathematician is greatly abridged 
by the adoption of suitable artifices. 



DIFFERENTIAL CO-EFFICIENTS. 51 

\\ -J— X ^ X 

Q. y z=l — ^^ — Multiply both numerator and de- 

nominator by the numerator, then 

x"^ — 2aj V 1 + ^'^. Whence y = Izj 

dy dy dz . 
dx dz dx 
1_ 1 

2 ~ 1 + 2x'— 2x Vf+^'' 

4^ — 2 vr+^"2 — 



Put 


z^l-\ 


and 




but 


dy 
dz 


and 


dz 




dx 



Vl+x'' 
2{lJr2x^-2x\/l-\-x'') . ^.^ 

dy_ 1 2(1^2x'-2xV'-r\^') 

dx~\^2x'-2x VlTf^ ^ VTT^^ 

2 



7. y = tan.-i 






Make — — ^^ --r- = 2; then ?/ = tan.~^2;, 

and ^ ^^^c?2 c??/ 



c?x dz dx dz \ -\- z'- 
1 ^ 1 a\a''-%x'f 



(^x «(a^ — 3.r-)- 

_ 3 (a^ + 2(1- X' + o;^) _ 3 (a- + a--)- 



a(a^— Sx'^)'^ a(a"^-3a;"0'* 



52 DIFFERENTIAL CALCULUS. 

therefore d^ _d^d^ _ a^{a^-Zx^y ^{a^ + x')\ 

dx " d^ dx ~ (a^ + x'^y ^ ^a'-Sx'J' 

dy 3a 

* * dx a^-\-x^' 

This is as it should be ; for 

Za^x — x' 



tan." 



therefore y = 3 tan."^ -• Make - = 2, then 

, . dy dy dz , 

y = 3tan. 2, and ^— = -^^ — y— ? 
^ ^ ax dz dx 

dy 3 3 3a2 . 

a- 



and 



(is 1^ , dy _ 3 a 



c?a; a ' ' dx a'^-\- x' 



* To prove this, take Eqs. 28 and 33 Plane Trig., and in 28 make 6 = 2a; 

then 

tan. a + tan. 2a 

tan. 3a = ; 

1 — tan. a tan. 2a 

Substituting in the second member of this the value of tan. 2a taken from Eq. 33, 
and reducing, we find 



1 — 3 tan. 2 a 

3a 2 z — X[ 



Dividing both numerator and denominator of the fraction 



a 3 — 3aa: ' 



3i_^ 
a a3 

X- 

a- 



hj a 3, it becomes — ^ ', and, comparing this with the formula for the 

1—3-^ 



tangent of 3a, we conclude that it is the expression for the tangent of three times 
the arc of which — is the tangent : therefore 

3tan-i-=tan.-i— ^^ ^=tan.-i ^^^2_3;,2) 



as was assumed. 



DIFFERENTIAL CO-EFFICIENTS. 53 

e«^(asin. flj — COS. ic) 



y^ 



a^+l 



^ _- . — e«^ (a sin. x — cos. x) 

dx a^ 4- 1 dx 

-$- e«^(a sin. ic — cos. x) = ae"''{a sin. x — cos. a?; 
dx ^ 





-\- e'"'{a cos. X + sin. ic 




=:(a2-|-l)e«^sin.ic; 


• • 


-^ = e^^sin. a:;. 


9. 


1 . ,6 + asin.a7 

V := sm. — , — z — ; . 


^ ^ a'^ ^ b'^ a + ^sin.ic 




& + asin.:c ,, 1 . , 


Put 


— —J — , — 2: Lliyu // — ,^- =- bin. *v, 

a + osin.o; ^a^—b^ 


and 


dy _ 1 c?y dz 
dx A^ a' — b'^ ^^ ^^ 


But 


dy _ 1 1 


dz Vl — 2-~ ^ ^6 + asin.icY 
N \(X -(- 6 sin.a?/ 


> 


a -j-&sin.a; 




s/ a'- — Z)"^ cos. X 



dz __{a^b sin. x) a cos. x — {l) - f a sin, a;) 5 cos , a; 
^"^ c/o; " " (a + rsm. oj)-^ 

(a- — V-^ COS. a? ^ 
" (a -|- /> sin. .i-)- 

. c??/ 1 a + ^ sin. x (a- — b'-) co^. x 

therefore -^ = -- — ^=:r , -^ ; , , , -^ ,n-2 

1 

a-\-b sin. x ' 



54 DIFFERENTIAL CALCULUS. 

10. If ?/ == cos -^ — ■-- , we should find, m 

like manner, 

^^ 1 

do; a-\-h COS. a; ' 

^ 1 / e^cos. ic 

11. ?/ = tan.~^ 



1 + e^sm. ic 



^ e^cos. a? ,, , , 

Put z = r— — —. ; then y — tan."-^ ^, 

1 + e sm. a? '^ ' 

dy dy dz 1 dz 

dx dzdx \ -\- z^ dx 
dz _ 
dx 
(e'^cos.o:; — e^sin.a:j)(14-e^sin.fl?) — e'^cos.cc(e^cos.a;-f-e^sm.£c) 

(1 + e^sin. a;)^ 

e^(cos. cc — sin. cc — e^) 

(l + e-^sin.ic)2 ' 

1 _ 1 _ (l + e^sin.x)2 



and 



1 + 3^ 1 I / e'^cos. cc V (1 + e^sin.x)^ + (e^cos.aj) 
\1 -|- e'^sin. xj 



(1 + e^sin. a?)^ 

d/it 
therefore 



12. y== 



1 + 2e^ sin. x -\- e^^ 

c??/ e^ (cos. a:; — sin. x — e^) 

c?a; "~" 1 + 2e-^ sin. cc + 6^"^ 

1 + cc dy_l — 2x — x'^ 

1+x-^' 5^~ (1 + «;')' * 



13. y — xlx, y~lJ^lx, 

7 ^ dy 2 

14. ?/ = ? cot. X, -f — -. — -— 

dx sm. 2x 

^r X dy a^ 

15. 2/~^ ^ 



VK - ^')' (^:r ~ (a'^ - ic^)^ 



DIFFERENTIAL CO-EFFICIENTS. 55 



(a/QC 



xX""" dv fx 



''■ '=[a ■ -£=% 1+^ 



ax 



18. 



x^ dy nx^~^ 



(1+^)"' dx {l+xf+^ 
19 g^-e-^ dy ^ 4 

20. 2/ = ^ (e" + e-% ^ = e- — ^2^ 

^ ^ ' ^' dx e-^-j-g-^* 



21. 2/ = (« + ^)'"(^ + ^)% 






1"PIT1 ^ fill 

22. y = — ^^ tan. a:^ + ^r, -/ = tan.* ic. 

3 dx 

^ + V'l — (T-^ 6?x Vl-a;-'^ (1 +2xVl - x'') 
24. 2/ = (a2 + ^2w^j^-i^ ^-^ = 2^tan.-i- + a. 



25. ?/ = M ?(a + ?;x-") 



ra-l 



c?y _ nbx 

dx ~ {a-^bx")f{a^hx"y 



Tt , X 



26. yz^/tan.f'^+-^), ^ = _J_. 

^ \4 2/' c^^ cos.o,' 

9Y __ ^^ 4" '^' dy j^ax — a 



28. 2/ 



JS 



a; (7// _ 



03' 



f?.^; (1 — .r) VI — ic"^ 






66 DIFFERENTIAL CALCULUS, 
\^l-\-x + Vl—x ^ 

X \" dy 






31. 2/ = 



l + Vl— ir/' c?x xVl — 

i dy __ xy 



X' 



32. ^/^tt^'^^--^^ ^ 



(a2 _ x')i 



■la. 



33. 2/ = eV^i±lY, '^^-,.. ^'-2 

V^^-*:/ rfx (a; + 1)^(^-1)1 

34_ ^_ y/T+^ + Vrr^ ' dy_ 2/ 1 

Vl + a^^-Vl — x^' <^-« ^A Vl-a; 

35. « = sin.-'-- '^^'^ ™ 



sm. •■ mx. 



(1 -m^a^^p 



36. y =ra?sm.-ia?^ -^ = sm.-^x + 



—1 

tow. X -a tan. x 



37. y~e ■ ~, -V- = e 



c?x Vl — iC^ 

dy ^ ~i 1 

dx 1 + aj2* 

38. y^x^^^~'^ ^^ - ^...-\. M^ + (1 - ^^) ^ sin.-^ a ;' 

39. yz= 



sm. x( 1 \ — 2>(x — sin.-i £c) cos. x. 

dy __ \ Vl- xV ^ ^ 

dx sin.^cc 

a-\-h tan. - , , 

. ^ _ , 2 dy oh 

7 , X CIX 9 J X 7 9 • 9 

a — otan.- a- cos.-- — 6- siu.- ~. 



_i 1 

41. y—x'^i dy = x^{l—lx) ^ 

dx x^ 



DIFFERENTIAL CO-EFFICIENTS. 57 



42. y^e'\ %^e'\\ 

ax 

43. 2/ = x% g = ^.V/l+?., + (?^)-. 



dx \x 

44. y = xS ^ = ,/e^l+fl-. 

ax X 



45. ?/ =r sin. 



46. y = tan.~i 



X dy 1 



47. 2/ = tan. VfZT^ 

^^ (cos.Vl-::^)^ 2v/l-^~ "TvT^l" 



48. 2/ = tan.~^(ntan. a;), 



^y_ 



49. . = (. + .)tan.-g^-V.., g^tan.-^^. 

50. 2/ = tan.-i_i^ dy__J(l-x^^) 



_i 2x dy 



51. ?/ = tan. ^ . _ 

52. 2/=.sin.-Vs^hr^ j! = Wr+T;;^ 



^x- 2 ^ ^ + <^'^>^ec. cc. 



53. y z=z sin.~^ 



aa? 



i + co; 



2> 



dy r?, (?> - r:c2) 

dx (b + co;*^) Vft^ +"( 267-="^^yS^Tp-^4' 



54. 2/ = \/r-.f'^sin.-^r-.r, 



dx v/i"3r;F 



58 DIFFERENTIAL CALCULUS. 

, xidji.h dii a Han. 5 1 

56. 2/ = sin.~^ -^ — 



Va"'— x"-' dx a' — x' {o? - x'- sec.^ h)\ 






V&-^-a2 



(52_^2)Va'^-x-2 



58. ^^tan.-i_ ^^-^^^-^ ^ ^ = 1. 



vr+ 



COS. X 



59. y-t..-iA"-^ -^')^^i"-^ \';^- K-^^)^ 

\ h-\r a COS. ^r / aj: a + i cos. x 

60. 2/ = sec. 1-^ ^ — 



2x'-V dx ^1-x' 

^1- 2/ = tan. n^ , dx~2{l^x^^y 

^ 1 — 0:^/2 + ^- l — x-'dx l+a;2 

63. 1^ (^+1)"^ L.tan.-i?i:;i-, in which 

f^{l-\-Sx-\- 3x^)i ^^ ^ 1 

64. From the equation 

. 72 + 1 . n 
sm — X sm. - ic 

sin. cc -[- sin. 2aj -}- . . . -j- sin. na:; = ~ ; 

sin. -ic 

Ji 

prove, by differentiating both members, that 

cos. x-\- 2 COS. 2x + 3 COS. Sx -{- . . . -\- n cos. nx is equal to 

w+1 . 1 . 2^+1 1 / . n+1 

sm. - X sm. X — - sm. — - — x 



2 2 2 2V 2 



sm.^ ^ X 



DIFFERENTIAL CO-EFFICIENTS. 59 

65. Admitting"^ that 

sm. aj sm ( — -\-x sm. — -{-x] . . . sm. { ■ n -\-x 

\m J \m ' J \ m . 

sin. mx 



2 wi — 1 ' 
Id which m is a positive whole number, prove that 

cot. ir + cot. ( — 4- X ) 4- . . . cot. ( Tt-\- x\z=imcot.mx. 

\m J ^ \ m J 

* As the equations assumed in this and the preceding example are not usually 
given in treatises on elementary trigonometry, they will be demonstrated in the 
Key to this work. 



SECTION IV. 

SUCCESSIYE DIFFERENTIAL CO-EFFICIENTS. 

48, The differential co-efficient of a function, f{x), of 
a single variable, being in general another function, /'(:c), 
of the same variable, we may subject this new function 
to the rules by which f^{x) was derived from /{x)^ and 
thus obtain the second derivative, or differential co-efficient, 
of the original function. The second differential co-efficient 
will, in turn, give rise to a third, and so on ; and we thus 
arrive at the successive derivatives, or differential co-efficients, 
of a function. 

The notation by which these successive differential co-effi- 
cients are indicated will be best explained by an example : — 

Let us take y =zx'^ ; then 

~^z=y^ ^= nx"~^ 1^* diff. co-efficient. 

^| = 2/'' = ^(^ — 1)^"~' • • 2*^ diff. co-efficient. 

, ^ :=: ^ (^) m^^ diff. co-efficient. 

These are the first, second, . . . m^^ differential co-efficients 
of the function y z=zf[x). It is sometimes convenient to de- 
note these by writing the function itself with as many dashes 
as there have been differentiations performed: thu3/'(:z;), 
f"{x)^ . . . f^'^^{x)^ are the first, second, . . . m^*" differ en- 

60 



DIFFERENTIAL CO-EFFICIENTS. Gl 

tial co-efficients of f{x), and have the same signification as 
ij/ y", . . . 2/(-\ 

In the example just given, it is evident, that, if n be a 
positive integer, the m*^ differential co-efficient will be inde- 
pendent of x^ that is, a constant, when m^:^ n ; and that the 
function will not have a differential co-efficient of a higher 
order than the n^^. In other cases and forms of function, 
there will be no limit to the number of differentiations that 
may be performed. 

The symbol. -^^, ^,... __|, are 

d^ dpi d^, 

. , ^ ^ dx dx^ rfa;"'-' , 

eqmvaJent to — ^ — ; — -j — j • • • — j — , and 

ax ctx CLx 

are read second, third, . . . m^^ differential co-efficient of y 
regarded as a function of x ; and are to be viewed as wholes, 
and not as fractions, having d^y, d^y, . . . d'^y, for their 
numerator, and dx'\ dx^, . . . dx"\ for their denominators: 
nor must the indices 2, 3, . . . m, be considered as exponents 
of powers, but as denoting the number of times the function 
has been differentiated. 

40, Successive differential co-efficients of the product of 
two functions of the same variable. Leibnitz' Theorem. 

Take u = yz, in which y and z are functions of .r; then, by 
Art. 20, we have 

dii __ dz dy . 

d~c~''^ dx'^^'dl' 
and, differentiating both members of this equation wKli rospoct 
to Xj we have 

d'^u __ ^2 d(j dz dy dz d" y 

dx' ~ ^ ^^ "^ du di: + dc dv + "^ Zi-- 

-^ dx'^ dx dx^" dx' 



62 DIFFERENTIAL CALCULUS. 

In like manner, we should find 

d^u d'^z dy d'^^z dz d'^y d^y 

dx^ ~ dx^ dx dx' dx dx'^ dx^ ' 

and 

d^u d^z dy d^z ^d'^it d^-z dz d^y d^v 

— y [- 4 — h 6 — - +4 4- z -' 

dx^ ^ dx^ ' dx dx^ dx''- dx^ ' dx dx^ ' dx^ 

This has been carried far enough to enable ns to discover 
inferentially the laws which govern the numerical co-efficients, 
and the indices of differentiation in the expressions for 

, ; — -. These laws are the same as those for the 

dx'- dx^ dx"^ 

co-efficients and exponents in the Binomial Formula ; for, in 

respect to differentiation, y may be regarded as y'^°\ and 



2 as s 



(0) 



To prove these laws to be general, let us assume them 
to hold when n is the index of differentiation. Then 
d^'u d^'z dy d^'-^z n — 1 d'^ y d"-'- z 

/ft I /yj I Y) - —I 

dx"" ~ ^ dx"" ~^ dx dx"-^ ^ 2 dx'^ dx"-- ^ * * * 

(71 — 1 ) {n — 2) . . . {n — r+^) d'-y d'^^'z 
+ ^^ 2.3. . .^ d^ d^ 

{n — l){n-2) . . . {n — r) dT^ d^'-^'^^^z 
2.3. . . r(r + iy " dx'''^'^ dx"'-^'-^^^ 



-\- n 



^ ^ dx^' 

Differentiating both members of this equation with respect 
to ic, reducing, and arranging the result, we find 

d^^^^~^db^^'^^'''^^dxd^'^ ' ' ' 

( 7^ + l)7^ . . . (7 ^4- l-r) d^ d—'z >^ 

^^' 2.3. . . (r"+l) dx'^^ dx''-' -t- • • -"^ dx^^^' 



DIFFERENTIAL CO-EFFICIENTS. 63 

Now, the laws of the co-efficients and indices in this devel- 
opment are the same as those assumed to be true in that 
from which it was immediately derived ; but, by actual opera- 
tion, we know them to hold when ?2 = 4 : they therefore hold 
when n = 5 ; and so on : hence they are universal. 

As an example of the above, take u = e^^y ; then, observing 

that-^^ = a'^e''^, Ave find 

dx^-^ V^ + ^'' dx^'^-Y-"" dx^-^-'-^-d^n 

Now, by examining the expression within the parentheses, 

. / d\^ 
we discover, that if ( a + ^ ) 2/ he developed by the Binomial 

Theorem, treating the symbol -y- as a quantity, and [ \y 

dV / dy , ^, 1 1 I cly dhj d^'y 

d-^y--- Ur '^^ '''''' '■'p^'"^'^ ""^L-d ■■■-dS' 

d" u 
we get that factor of the development of -j-~ : hence 

cix 

ITx'^ ~ ~^^~ ~ ^"' \^ + Tx) ^ 

is a convenient and abridged form of writing the ?z'''' differen- 
tial co-efficient of the function u = e^-^y, 

50, If n be a positive whole number, we may prove that 

d"u _d"icv _ d''-^ / dv\ n(n—\)d"-" / d''v\ 
^ d^'^ ~ dx^ ^ 'dx^-^ \ dx) "^ 1.2 dx'^-^ y^' dx"') 

_<fec. + <fec. . . . +(-1)",,^. . . ^1). 

For let y=zuv, in which both lo and v are I'unctions of x : 
then, differentiating witli respect io x, wo have 

df/ d.fiv dr , (/// 

dx dx dx dx 



64 DIFFERENTIAL CALCULUS. 

, du d. uv dv 

whence 2; -— i= — u-=-; 

ax ax dx 

and the theorem holds when 7i = 1. 

Now, differentiate both members of Eq. 1 ; then 

d'^'^^u , dvd^'u d^+'^uv d"" f dv 

qj _l ■ IVi I ql 

dx'''^'^ dx dx'' dx""-^^ dx'''\ dx 

^(^_l)(i«-i / dH 



"^ 2 c?x"-^ \ dx'- 



(2). 



If the theorem holds for y z=zuv when the index n has some 

dv 
assigned value, it will also hold for u -, when n has the same 

dv 
value. Changing, in Eq. 1, v into , we have 

dv d^'u ^d"" / dv\ cZ"-i / d'-v 

'dx dx"" ~ dx^ \ dx) ~~ ^ dx''-^ \ dx^ 

n{n — 1) d''-^ / d'^v 
^ "~T:2 'cL^^^ y dx\ 

_&c. + ...+ (-l)''«^^^-,. ..(3). 

Subtracting Eq. 3 from 2, member from member, and redu- 
cing, we find 

^ dx'^^ ~ "^^+1" ~ ("^ + ^^ cZ:^ \^dxj 

. , ... d--' ( d'v\ , , 

Hence, if the theorem is true for any assigned value of n, 
it is true when the index is w + 1- It is true when n^=l] it 



DIFFERENTIAL CO-EFFICIENTS. 65 

is therefore true when ?^ z= 2 ; and so on ; that is, it is univer- 
sally true. 

Examples. 

1 v = ?^^^=^^=- — 

' dx x' dx'^ x^ 

d^y _l^2 d^y __ _ 1.2.3 

dx^ ~ x^^ dx^ ~ x^ 

dx"" ' ^ x"" ' 

. dy . / Tc\ 

2. V = sin. X, — = COS. X = sm. [ x A — , 
^ ' dx V 2/ 

d sin. \ x-\- - , 
dHj \^2) ,A . ( , 2^' 

-z— T -=^ = COS. \ X A — =1 sm. [ X A 

dx' dx \ ^2/ \ 2 

7 • / 27?\ 

a sm. { X A 

d'y \ ^ 2 ; / 2^ . / ^3;r\ 

■ -T-T= ' — ~ — ; = COS. cc -|- -- = sm. [ x -{---- ], 

dx^ dx \ ~ 2 J \ ~ 2 / 



d^y / n7t\ 

—=— = sm. { X A- -~ ] 

dx'' \ 2 / 

dy . / , ^' 

dx 



3. y z= COS. X, ~ ^= — sin. x = cos. ix-\--\, 



d"7j f 7in\ 

— = COS. \ X -\- . 

dx"" \ ~ 2 I 

4. y z=z COS. ax, -z~ = a" cos. f ax + - ). 

5. y =z tan. ;?^ -f- sec. x, 

dy _ I sfn. 0? _^ 1 + s'in. x 1 

dx cos.-.f cos.-O/' cos.-.u 1 — sin. :r^ 

d'^y __ cos. a; 

cZj;' (1 — sin. a?)"^ 





66 DIFFERENTIAL CALCULUS. 

3 sin. oj — sin. Zx 



6. ?/ =: sm.^a; =r 

d^'y 3 . / . nn^ 3" . /^ , titt^ 

— — = - sm. [ X -\- ~- ] — -- sm. [6x-\--- 

^ ' dx"" X 

10. 2/ = («^ + a^)taB-^, S = r:r^"' 



a dx^ {a^ + x^)^ 
11. y=ze~^co8.x, 7~T == — 4<?~'^cos. a;; hence 

ClX 











zrO. 






12. 


y = 


1-0? 

'l + o;' 


d^'y _ 
dx"" ■ 


= ( — 


'^ ^1 + ^)^ 




13. 


y^ 


:(e- + e-)«, 












d'y 

dx'~ 


:^2(gx+e— ) 


« _ 4^^ (^ _ 


.1)(6 


^ + e-^)"-2. 




14. 


y'^ 


= sec. 2a?, 




= 3r 


. 




15. 


y = 


ax -{-h 
'x'-c^' 












d-y _ 
dx"" 


.(-.)■-;: 


. nl b-\-ac 


h — ac \ 




16 


2/ = 


1 












cc'^ - a'' 




d-y 
dx"" 


z=(- 


1.2.3... 
^ 2a 




(n+l) 


— (aj + a)-^'^+i^ 
/ 



DIFFERENTIAL CO-EFFICIENTS. 67 



17. y ^zx"" ^m.x, 
^-1.2.3. . . n -j sin.aj + -^sm.f^ + - 

.n{n-l) 27t 

n{n-l){n-2) 3 . / , 3;r\ , , 

+ -xm^:!-^ ^^^^- (^ + Y ) + ^^• 

18. ^ = tan.-^, ^-cos.^^, 
a a ax a 



l=^«os-^ + ^ cos.'S 






rf«^ o V a ^2/ a 



rfa!''-a^''°'-^^"^^2y^"- a' 



Because tan.-^^ zzz ^ - tan.-i ^, make tan.-^ ^ -^ d - then 
« 2 a' a 2 



= ( — l)"-i sin. 7279 ; also cos. '^ = 



^^ (a'^ + a;2)^' 



COS." -^ = 



/7 " 

(a^ + x^Y 



Substituting tlicso values in the expression for — ^ wo huvo 

dx'' ' 

(a- + uj2), 



68 DIFFERENTIAL CALCULUS. 

19. Prove that 

^" / 1 \ _(- If 1.2.3. . . n sin. (n + l)^ 



7 1 V(-^ 



(a^ 4" ^ ) 
Proceed as follows : — 

6/tan.~^- ^'^ / 1 \ 1 ^""^^tan.-^- 



dx ~a' + x' dx''\a''^x''J a dx'' + ^ 

and, from this point, the process is similar to that pursued in 
the preceding example. 
20. Prove that 

d"" / X \ _ (— If 1.2.3. . . n COS. ( n -{- 1' 

dx^ [a:^ ' ~~ 

By Art, 49, 



^^n \ r/-i _\_ rp-l j n-\-\ 

ax \a -Yx J ^^2 ^ ^2)^2- 



d"" ( x \ d"" / 1 \ , d''-'^ ( 1 



and finding the value of each term in second member, as in 
last example, we get 

c?" / ^ \ _ (— 1)^ 1.2. 3. .. 71 sin, (n + l)^ 

'dx- V^M^^ V ~ "" ■ . „ , ,"±) 

^ / a{a- -{- x^) ^ 

(— 1)"-^ 1.2.3. . . n sin. nd 

aia'^ + x'-y 
_ (- If 1. 2. 3. . . 71 COS. {n-\-l)d 

21. When 2/ = sin. (m sin.~^ a:;), prove that 

22. When y ^^a cos. Ix -\- b sin. /x, prove that 

also that 

a;' ^ J + (2h + 1) :c V-n! + («' + 1) 3-r = 0- 



SECTION V. 

EELATIONS EXISTING BETWEEN REAL FUNCTIONS OP A SINGLE 
VARIABLE AND THEIR DIFFERENTIAL CO-EFFICIENTS. TAY- 
LOR'S THEOREM. MACLAURIN'S THEOREM. 

31, If the increment ax of the variable x in the function 
y ^ F{x) produces the increment Mj of y, then the ratio 

-- , having for its hmit ^ =: F\x), wiU, before reaching this 

A X CCX 

limit, and when ax is very small, take the sign of F^ (x) ; that 
is, it will be positive if the differential co-efiicient is positive, 
and negative if the differential co-efficient is negative. But, 
when a ratio is positive, its terms have the same sign ; and 
when negative, they have opposite signs. Hence, if the dil- 
ferential co-efficient of a function be positive, the function will 
increase or decrease according as the variable increases or de- 
creases; but, if the differential co-efficient be negative, the func- 
tion will decrease as the variable increases, and the opposite. 

32. Suppose that the function y = i^(-r) is continuous be- 
tween the limits answering to the assigned values .r =i .i\,, 
x = x^, of the variable, and that the variable passes by insen- 
sible degrees from tlio first to the second of these values : 
then, by the foregoing article, the function cannot change 
from an increasing to a decreasing, or (Vom a decreasing to an 
increasing function, unless the (lilVcrvMidal co-elliciont changes 
its sign from positive to negative, or iVoni negative io positive. 
But a function can change its sign only when it passes 

GO 



70 DIFFERENTIAL CALCULUS. 

through zero or infinity. If continuous, the change of sign 
will occur in passing through zero ; if discontinuous, in pass- 
ing through infinity. 

53, If the function y^=.F{x) vanishes for x^^x^^ and is 
continuous for values of x which are indefinitely near x^^ 

then 

F{x^ -}- Acc) — LxF' [x^) + ^xr . . . (Art. 15), 

where r is a very small quantity when A a? is very small. The 
sign of the second member will therefore be determined by 
that Qi F' {x)\ hence, if x = ccq -}- Aic differs very little from x^^ 
jP(iCo + Aa:) = i^ (x) > if F< {x) > 0, 
F {x^^ Lx) — F {x) ^^ if i^^(x)<0. 
64:. Suppose that the two functions F {x)^f {x)^2.xq real, 
and that they, as well as their differential co-efficients, are 
continuous between the limits, answering to the values x^ and 
Xy-^h of the variable ; suppose also, that, between these limits, 
f {x) does not undergo a change of sign ; that is, for the in- 
termediate values of x^ f{^) must constantly be either an 
increasing or a decreasing function : then the ratio of the 

differences 

F{x, + h) - F{x,), fix, + h) -fix,), 

will be equal to that of the derivatives F^ (x), f (a?), when in 
these X has some value between x, and x, +/i; that is, if d^ 
be a proper fraction, we shall have 

Fiyx, + li)—F{x, ) _ F' (^1 -h d^li ) 

To prove this, let A be the least and B the greatest algebraic 

values that the fraction ■ ^ / can have for values of x be- 

tween x^^ and x^-^li ; then the two differences, 
F'\x) _ F_^(x) _ 
/'(x) ^' f\x) ^' 



DIFFERENTIAL CO-EFFICIENTS. 71 

must have opposite signs for any of these values of x; and the 
same will be true for 

F' {X) - Af {x), F' {x) - Bf {x), , 
because, by hypothesis, f {x) has an invariable sign between 
its limiting values. But these last expressions are the differ- 
ential co-efficients of the two functions 

F{x)-Af{x),F{x)-Bf{x). 
One of these functions, therefore (Art. 52), must be constantly 
increasing, and the other constantly decreasing, Avhile the 
values of x are limited by Xi and Xi-\-h. 

If, then, the value answering to Xi be subtracted from that 
answering to ccj + A for the one and the other, we have the 
two expressions, 

F{x, + h) - F{x, ) - A (f{x, + li) -f{xS), 

F{x, + h) - F{x,) - B {f{x, + K) -f{xS), 

one of which must be positive, and the other negative. 
Wherefore it follows, that, if both be divided by/(^i + ^) 
— /(Xi), the quotients 

F{x, + h)-F {x,) _ 

f{x, + h)-f{x{) 

F{x, + li)-F{ x,) _ 

., • ,1 , . F{x,-^h)-F{x,) . 
have opposite SI i^^ns ; tliat is, — . , ., , ,. -. is c:roater 

than A, and less than B, and is therefore comin-isod betwoon 

F' ix) 
these p-rcatost and least values of — ,- v- But F' ( .r) and f i .r") 
^ J\x) ^ 

being continuous, whiU> x })assos by insonslblo gradations from 
x-i to X, -\-h, the ratio -^^v must pass tliroiiuli all values iu- 



72 DIFFERENTIAL CALCULUS. 

termediate to its greatest and least values. Hence there must 
be some value of x between Xi ando^i + h that will render 
the ratio of the differential co-efiicients equal to the ratio of 
the differences of the functions. 

Let ^1 be a variable proper fraction: then, from what pre- 
cedeSj a value may be assigned it, that, agreeably to our enun- 
ciation, will cause it to satisfy the equation 

F {x, + h)-F{x^) ^ F'{x,-\-d,h) 

dS, It has been assumed in what precedes that /^ (x) re- 
tains the same sign between the initial and final values of a;; 
but the proposition is true when the assumption is made with 
reference to F' {x)^ instead oi f {x). For, if jP'(:i;) does not 
change its sign, by the same course of reasoning we can prove 

that 

f{x,-\-h)-f{x,) f'{x, + d,h) 



whence 



F{x,-^h)-F{x{) F (x, + d,hy 
F(x, + 7i)-F(x,) F'(x,^d,h) 



J\x, + h)-J\x,) ~f'{x,-\-d,h) 

SO, From the theorem established in Art. 54, we deduce 
the following consequences: — 

1st, If F{x) and /(x) both become zero for the particular 
value X =: x^, then 

F{x,^ h)_ F^{x,-^ d,h) 

7{x, + h) r{x,-\-0,li) •'•^''^' 

2d, If the differential co-efficients up to the (?^ — 1) *^ order 
of both F(x) andy (x) vanish for x = Xi, those of the second 
being constantly positive or constantly negative between the 
limits corresponding to x == x^, x ^= x^-]- h, while the func- 
tions themselves do not vanish for this particular value of ic^ 



DIFFERENTIAL CO-EFFICIENTS. 73 

then, from what has just been proved, we shall have the fol- 
lowing relations : — 

f'{x,-\-d,h) r{x,+d,h) 

F" (iCi + ^2 ll) _ F"'(y X^^d^ll) 



/^"-^H^i + ^.-i^O /^"^ (^1 + ^. /O ' 

therefore 

F{x,-\-h)-F{ x,) ^ F^n^{x, + Oji) 
/{x, + h)-/(x,) fo^x,Jr(^nh) 

Since, in the reasoning, no condition has been imposed on 0„, 
except that it be a proper fraction, we may omit the subscript 
Uf and thus have 

/{-ci + h) -/(x,) - /»' (x, + eh) ^"i- 

If the functions reduce to zero, with their derivatives for 
X zzz Xj, we have 

Fix^+h)^F--^(x, + Oh) 

/(^i + /') f">{x,-\-Oh) ^ ''• 

Making the further supposition that a^^ = 0, then 

F[Ji) ^F^^'^iOh)^ 

but, because this is true for any value of//, x may bo written 
for 7i ; and thus 

F {x) _ F'"' {Oa') 
/(x)-/^"\Ox) ^'^' 



in], The conditions relative to /\x) that liavo boon ini- 

10 



74 DIFFERENTIAL CALCULUS. 

posed in the preceding j)ropositions are satisfied when f{x) 
= (a; — Xi)^ : whence 

f" i^x) == ?i (?i — 1) (a? — iri)"-2 



/^)(x) = 1.2.3. . . n=.f^\x^-\-dli). 
Here/(ic) and its successive derivatives, up to the (n— 1)*^, 
vanish for xz=zx^\ and since, in Eq. h^ the denominator of the 
first member, 

/(•^i + ^ -f{^i) = {^i - ^1 + A)" - (^1 — x,)^' = h% 
we haA^e 

F{x, + h) - J'(xi) = ^^^l°_,^^ F'-"> {X, + Oh). 

When n = 1, this gives 

F{x, + A) — i^(xi) = liF' (x^ + ^A). 
If i^(xi) == as well as/(xi) = 0, then 

^^"^^ + ^'^ = \ A'. ..,/ "" ^^^ + *'')• 

Making x^ = 0, and then, writing x for A in the preceding 
equations, they become 

i^^") {dx) 



F{x)- 


-^(0) = 


"1.2.3. . . n 


F{x)- 


■^(0)- 


-.xF'{dx) 




i^(a.) = 


^" 




■ 1.2.3. . . n 


iQuatiou 


i^w 


_i^(")(/9j^) 



i^^") {Ox). 

^. ^ _ ^ ,^^ ^ (Eq. d, Art. .56), ex- 

pressed in words, enunciates the following theorem: viz.: If 
there be two functions, F{x)^/{x), which, Avith their difi'eren- 
tial co-efficients, are continuous, and which, with these differ- 



DIFFERENTIAL CO-EFFICIENTS. 75 

ential co-efficients up to the (n — 1)^^ order inclusively, vanish 
for X = ; and if, further, the first n differential co-efficients 
of one of these functions are constantly of the same sign for 
values of the variable between zero and another assigned 
value ; then the ratio of the functions will be equal to that of 
the 71 "' differential co-efficients, when, in the latter, some inter- 
mediate value is given to the variable. 

The importance of this theorem warrants us in giving it an 
independent demonstration. 

Let F{x) andy(jc) be two functions which vanish for x ^ {)] 
and suppose, first, that the differential co-efficient f {x) of the 
second does not vanish for this value of the variable, and that 
it retains constantly the same sign between x = and x i= 7i, 
which requires that/(x) be continually increasing, or continu- 
ally decreasing, between these limits (xlrt. 51), and therefore 
constantly positive or constantly negative, since f{x) =z 
when X =^ 0', and let A denote the least and B the greatest of 

FUx) 
the values assumed by the ratio ^, ; / for values of x be- 

f (^) 

FUx) 
tween zero and h: then the two quantities, — — ^ — A, 

F' (x) . 

-j.,^ ' — -5, will have opposite signs; and, since f (.r) does not 

change sign, the same will be true of the differences, 

F' {x) -Af {x), F^ {x) - Br (x): 

but these last are the diHerential co-efficients of the two 
functions, 

F\x)-A/\x),F^x)-Ji/\x), 

one of Avhich (Art. 51) nnist tluMvCoro bo constantly in- 
creasing, and the otluu' constanllv (K\'roasing: that is. since 
both F(x) andy(.r) vanish lor x — 0, one must be constantly 



76 > DIFFERENTIAL CALCULUS. 

positive and the other coDstantly negative between the limits 
answering to x ^^^, x^^ li. Therefore^ because y(;r) is of in- 
variable sign, 

/(^) /(^) ' /(•^) /(^) 

are of opposite signs : whence it follows that the ratio of the 
functions is comprised between the least and the greatest 
values of the ratio of the differential co-efficients. But, if the 
variable be made to pass by insensible degrees from to h, 

. F' (x) , . 

the ratio — -, which is by hypothesis continuous, must pass 

J \^) 
through all values intermediate to A and B. If then d denote 
a proper fraction, it w^ill admit of a value such that the equation 

Fix) F' (dx) .„ , . n . 

— ^ — = ^ — - will be satisned. 

f{x) /' (te) 

If the differential co-efficients of both functions, from the 

1st to the (72 — 1)*^' orders inclusively, vanish for a? = 0, by 

reasoning upon them as we have upon the functions, we have 

F'{d^x) __F" (d.^x) _F"'{d^x) _ _ F^"\dx) ^ 



/'(Oi 


X) 


f" 


[^d^x) 


f" 


[ji,x) 


whence 






F{x)^ 


_i^( 


")(^x) 



58, It is to be observed that the only conditions upon which 
the equations 

i^(^i + 70 - i^(^i)= liF' (^1 + dli), 
F{x)— F{S)) = xF'{dx), 

depend, are, that F(x), and its differential co-efficients up to the 
order involved in the equations, should be continuous between 
the assigned limits of the variable. 



DIFFERENTIAL CO-EFFICIENTS. 77 

59. From the equation F{xy + h) — F{x{) = hF'{xy -\- Oh) 
of Art. 56, it may be shown, that, if the differential co-effi- 
cient with respect to x of any expression is zero for all values 
of a:, such expression is independent of x ; for, if F' [x] is zero 
for all values of x, the above equation becomes 

F{x^ + h) — F{x^) = ; or, F{x^ + h) = F{x{) ; 

that is, the function does not vary with, and is therefore in- 
dependent of, X. It is plain, that, if the differential co-efficient 
is not equal to zero, the expression will vary with x. Hence 
those expressions only are independent of a variable for which 
the differential co-efficients with respect to that variable are 
zero for all values of the variable. And further : if two func- 
tions have the same differential co-efficient with respect to any 
variable, such functions can differ only by a constant; for 
the differential co-efficient of the function which is the differ- 
ence of these functions is zero by hypothesis : therefore, by 
what precedes, this difference must be independent of x; that 
is, constant. 

GO, Suppose F{x) to be real and continuous ; then, by 
means of the foregoing principles, we may find the develop- 
ment of this function arranged according to the ascending 
positive powers of x. 

For we have. Art. 50, 

F{x) — F(0) = xF^ (Ox) = xF' (0) + 7?i .c 
by making F' (Ox) = F' [0] + li^ ; 

whence 

F{x) - F{0) - xF' (0) = 7?i X : 

from which it is seen that li^x is a (piantity that reduces to 
zero when a; is zero; and tlie same is true of F' {x) — F\^^, 
which is its first derivative witli respect to x. Its second doriv- 



78 



DIFFERENTIAL CALCULUS. 



ative is F" [x). Wherefore^ by the article already referred to, 

F{x) - i^(0) - xF'{^) = BiX = -^ F"{dx). 
Making 

then, as before, 

F{x)-F{^)-xF'{^)-^^ F"{0) = R, r-- 



F"{dx) = F"{^)^R.,, 

^2 



X' 



and it is evident that J? 2 -zr—^ is a quantity, which, with its first 

1. Z 

and second derivatives, 

F{x) — F' (0) — xF" (0), F" {x) —F" (0), 
vanishes with ic, and that its third derivative is F"'{x) : there- 
fore we have 



F{x) - i^(0) - xF\^) - —- F"{^) 



72-3 ^"'(^^)- 



X" 



F^^\dx). 



Next, place F"'{Qx) ■=. F"'{^) + itg, and then proceed as be- 

R x^ 
fore, and so on, bearing in mind that the expressions — ^ — , 

R x^ 
^ . . . , together with their derivatives up to the (n — iy^ 

order inclusively, vanish for a? = ; and we should have, for 
our final result, 
F{x)-F{0)-xF'(0)- ^F"iO)- 

1.2. .(72-1) ^^ 

whence 

F{x) = F{0)+xF^iO)+^F^^iO) 

^ 1.2. ..{^71-1) ^^^ 1.2.3. . .n ^^ 

And it appears that any real and continuous function F{x) of 
x is composed of the part 

F(O) + xF'(0)+^J"'(O) + ... 



+ 



X 



n-l 



1.2.3. . . (n-l) 



jr('^-i)(0) 



DIFFERENTIAL CO-EFFICIENTS. 79 

which is entire and rational in respect to x, and of the re- 
mainder 

If the function is entire, and of the 7i'^ degree, its deriva- 
tive of the n^^ order will be a constant; that is, F^"^^ {dx) 
= i^" (0); in which case the development will terminate with 
the {n-\- \y' term, and the remainder will be zero. For ex- 
ample, if i^(a?) = (1 + xYj we should have 

n — \ 

(1 + xY — l-\-nx-^n — - — - iT^ + • ■ • + nx""'^ + x"". 

A 

6l» The same principles also enable us to find the devel- 
opment of F{x-{-7i), arranged according to the ascending- 
powers of either x or h. For we have, Art. 5Q, 
F{x+'h)-F(x)=7iF'{x-{- dji). 
Make 7iF^ {x + dji) =z hF' {x) + R^, 

then 

F(^x + h) — F{x) — 7iF'{x) = R,, 

From this it is seen that B^ is a function of both x and Ji, and 
that it, and also its first derivative F^ (x -\- h) — F^ (x) with 
respect to 7i, will vanish for /i nr 0: hence (Art. 56) 

Fi^x + /O - F{x) - hF{x) = !^^^~F"{x + dJi) 

= {'^^F^'{x)+Ii, 

bv making 

F" {x-}-Oji) = F" {x) + R. : 
whence 

F{x + h) - F{x) - hF{x) — ^^\^ F'{x) = R, ; 

and R.,, together with its first and second derivatives with 
respect to h, will vanish for // == : therefore 

A- .... - h' 






^/l). 



80 DIFFERENTIAL CALCULUS. 

The manner of carrying on these operations is sufficiently 
obvious. We may write 

F{x -Y h) - F{x) - liF' {x)^ 

_i^ F'(X)- ... hn 

1.2 ^^ ^^^ir(«)(^ + ,;,): 

- 1.2.3.. ■(.- T)^'"-"(-) 
whence 

i7(a: + /,) = F{x) + AF'(a;) + A!. F"(«) + . . . 

If, in this last equation, we first make cc = 0, and then, in the 
result, write x for /i, we find 

F{x) = F(0) + xF'(0) + ^ ^-"(0) + . . . 

from which it appears that the formula of Art. 60 is but a 
particular case of that just established. But formula 1 may 
also be deduced from 2. For, in 2, change x into h, and make 
F{h) =/(a; + //) ; then, taking the derivatives with respect to 
A, and in the results making h == 0, we have 

F'{h)=f{x-^h), 

F"{li) =f"{x + A) . . . F^"\Oli) =/") {x + dli), 
F^{0)=f{x), 

F'\0) =f' {x) . . . i^(«) {dh) =f'"^ {x + 6h). 
But, when h takes the place of x, Eq. 2 becomes 

F{h) = F{0) + hF(0) + . . . + 3- ^3"^^^^ - i^-^ (^/O ; 

and in this, substituting the values of F{7i), F(0),F'{0). . . 
F"{6h), we have 



DIFFERENTIAL CO-EFFICIENTS. 81 

f{x + h) =f{x) + hf'{x) + ^{-;^f"{x) + . . . 

which agrees with formula 1. 

62, When F{x) is such that the expression 

i^("> {Ox) 



1.2.3. . .n 

for values of x between assigned limits continually decreases 
as n increases, then, by making n = oo, the formula, 

Fix) = F(0) + xF'(0) + -^i^"(0) + . . . 

of Art. 61 will give rise to a converging series ; and it may be 
written 

Fix) = FiO)+xFiO) + ^F'iO) 

which is Maclaurin's Formula. 

So also if F{x) is such that the expression 

for values of x between assigned limits continually decreases 
as n increases, then, making n =z oo^ the formula. 

Fix + h) = Fix) + !<F(.v) + ^^ F"ix) 

of the preceding article may bo written 

F^,. + /,) = Fix) + JiFix) + ^''' F'ix) 

+ r^^'"" (■<•) + ■•• (-\ 

1. -J. i> 

which is Taylor's Fornuda. 
11 



82 DIFFERENTIAL CALCULUS. 

63. It may be shown that the quantities, 

1.2.3. . .n ' rXS. . . n' 
become zero when n is infinite. For take the expression 

n -A- 1 

m{n — m -\- \) = m {n -^ 1) — m^ = 2m — ^ m^ 

A 

n + 1\2 /;^ 4- l\ 2 n-^l ^ 

4^)-(-t~-)+'^^-2 '' 



2 / \ 2 
which last form shows that the product m{n — m -\-\) in- 

0% 4-1 
creases as m increases from 1 to — - — ; that is, the product 

A 

increases as the factors approach equahty. This is also shown 

/n-\-\ \ 

by the differential co-efficient 2 i — — — m j of the product 

taken with respect to m. Giving to m, in succession, the 
values Ij 2, 3, . . . n, the product will assume the successive 
values 

n, 2(^-1), 3(?i — 2). . . (?^- 2)3, {n — l)"!, n, 

which increase from n up to a certain limit, and then decrease 
by the same gradations down to n again. 

As n is the least value that this product assumes, the con- 
tinued product of these results, of which there are n, will be 
greater than n^ ; that is, 

n. 2(n— 1). 3(?i— 2). . . (?i— 2)3. (?^— 1)2. n 

= fl. 2. 3. . . {n — 1) nX > n"; 

whence 

1.2.3. . .(n-l)n:>J: .'.:r^ < (^4" 

^ ^ ^ 1.2.3. .. n ^^ \\/n 



DIFFERENTIAL CO-EFFICIENTS. 83 



X 



But, if X is finite, ( ) will be zero when n is infinite : hence 



X' 



and the same is true of 



1.2.3. . .n 

in 



3Z when n = go 



1.2.3. . .n 

Therefore it follows, that if F^") ( dx), F^^^ {x + dli), are finite, 
the products 



1.2.3. . . n ^ ' 1.2.3. . . 72 

will diminish without limit as n is made to increase without 
limit; and we can, in such cases, employ Maclaurin's Formula 
for the development of i^(ic), and that of Taylor for the develop- 
ment of i^(x-|-^), into series arranged according to the ascend- 
ing powers of x for the first, and of either x or li for the second. 

G4z. Maclaurin's Theorem, when applicable, may be stated 
as follows : The first term of the development of F{^x) is 
what the function becomes when a? = ; the second term is x 
multiplied by what the first differential co-efficient of the 
function becomes when x = ; the third term is the second 
power of X divided by 1 X 2, and this quotient multiplied by 
w^hatthe second differential co-efficient of the function becomes 
when iT =1 ; and the {ii-\-\y^\ or general term, is the n^^ 
power of X divided by the product of the natural numbers 
from 1 to n inclusive, and this quotient multiplied by what 
the 71 '^' differential co-efficient of the function becomes when 
x — ^. 

This theorem is of very general application for the expan- 
sion of functions of single variables, examples k^K which will be 
shortly given ; but it is by no means universal : lor 

y = /.r, y =z cot. .r, y =: a' , 



84 DIFFERENTIAL CALCULUS. 

are functions which become infinite when a: = ; and hence 
the first term in Maclaurin's Formula would be infinite, while 
the function for other values of x would be finite. There are 
other functions, such as ?/ = ax^, for which, though the func- 
tions themselves remain finite for x = 0, their first, or some 
of the following difi'erential co-eflScients, become infinite for 
this value of the variable ; and, in such cases also, the for- 
mula would fail to give the development of the functions. 

Go, Taylor's Theorem may be enunciated as follows: 
When a function F{x -[- h) of the algebraic sum of two varia- 
bles can be developed into a series arranged according to the 
ascending powers of either taken as the leading variable, the 
first term is what the function becomes when this variable is 
made equal to zero ; the second term is the first power of the 
leading variable multiplied by the first differential co-efficient 
of the first term taken with respect to the other variable ; 
the third term is the second power of the leading variable 
divided by 1x2, and this quotient multiplied by the second 
differential co-efficient of the first term; and the {n + 1)'\ or 
general term, is the n^^ power of the leading variable divided 
by the product of the natural numbers from 1 to n inclusive, 
and this quotient multiplied by the 7i^^ differential co-efficient 
of the first term. 

66. In Taylor's Formula, the co-efficients of the different 
powers of the leading variable are functions of the other 
variable. When one or more of these functions are such, that, 
for a particular value of the second variable, they become in- 
finite, the formula fails to give the development of the origi- 
nal function for that value of the second variable ; for then 
the function ceases to depend on the second variable, and is a 



DIFFERENTIAL CO-EFFICIENTS. 85 

function of the first variable alone, and will not necessarily be 
infinite for the assigned value of the second variable. 
For example, if we have 

F{x) = s/x^^a, 
then F{x + A) = ^{x — a + h). ' 

"When oj = a, F{x) =: 0, and the first and all the higher 
differential co-efficients of F(x) become infinite for this partic- 
ular value of x ; while, for this value. F{x -\- h) = y'A. 

It will be observed that there is a marked difference be- 
tween the failing cases for Maclaurin's and Taylor's Theorems. 
When Maclaurin's fails for one value of the variable (x = 0), 
it fails for all ; whereas Taylor's may fail for one value of the 
second variable, but give the true development of the function 
for all other values. 

07. If a function becomes infinite for a finite value of the 
variable, its differential co-efficient will be infinite at the same 
time. In the case of an algebraic function, this follows from 
the fact that such function can become infinite for a finite 
value of the variable, only when it is in the form of a fraction 
wdiose denominator reduces to zero. But the denominator of a 
fraction never disappears in the process of differentiation : 
hence, if the function has a vanishing denominator, so will its 
differential co-efficient. In the case of transcendental fnnt'- 
tions, it is only by the examination of the diiforent forms tliat 
the truth of this proposition can be establislied. Thus, in the 
logarithmic function ij = Ix, y becomes infinite for x — : 

- = is also infinite for this value of x: and for the expo- 
dx X 

nential function y =r a'', which, if a > I, becomes infniile when 



86 DIFFERENTIAL CALCULUS. 

X = 0, the differential co-efficient is -^ = ^ a ^, which is 

ax X- 

infinite when ic ^ 0. 

The circular functions tan. x^ cot. x^ sec. x^ cosec. x^ which 
may become infinite for finite values of x, when expressed in 
terms of sin. x^ cos. x^ are fractional forms to which the reason- 
ing in reference to algebraic functions applies. 

If a function becomes infinite for an infinite value of the 
variable, it does not follow that the differential co-efficient 
becomes infinite at the same time. 

Thus, in the example y z=zlx^ -^ = ^^ and y is infinite when 

iJjX X 

ic = oo ; but — = for this value of x. 

68, It was remarked in Art. 62, that, unless F{x) and 
F(x -\- li) are such that 

F{x)+nF'{x) + ~F"{x)+..., 

give rise to converging series, the formulas of Maclaurin and 
Taylor will not serve for the expansion of these functions. 

A series in general is a succession of quantities any one of 
which is derived, according to a fixed law, from one or more 
of those which precede it. If Uq,Ui,ic2,U2, . . . u^, are such 
quantities called the terms of the series, then we have 

S« = ^^o + ^l^-^2 + ^3 + • • -^^z-i 
for the sum of the first n terms. When this sum approaches 
indefinitely a finite limit S, as n continually increases, the 
series is said to be converging, and the limit' in question is 
called the sum of the series ; but, if the sum jS„ does not thus 



DIFFERENTIAL CO-EFFICIENTS. 87 

approach any fixed limit as n increases indefinitely, the series 
is said to be diverging, and has no sum. 
The geometrical series 

05, aVj ar'^^ . . . ar""^ 

having ar^ for its general term, has for its sum 

1 —r"" a ar"" 



» V' ' ^ 1 — r 1 — rl—r 

It is evident that, as n increases^ this sum converges towards 

the fixed limit if r is less than 1 ; and that, on the con- 

1 — r 

trary, as n increases, the sum also increases indefinitely if r is 

greater than 1. 

We are assured of the convergence of the series 

when, as n increases, the sum 

>S'^ = ^0 + ^1 +^2+ • • • '^('n-l 

converges to a fixed limit S, and when, at the same time, the 
differences 

Snj^-^ — 8n = U,,, Sf,_^2 — ^>i = '^f'n + ^^m + 1 ' ' ' , 

vanish when n is made infinite. 

The limits assigned this work do not permit an investiga- 
tion of the rules by which, in many cases, the convergence or 
divergence of a series may be ascertained. 

60, Admitting that F{x) can bo expanded into a series 
arranged according to tlie ascending integral powers of .r, 
Maclaurin's Theorem may bo demonstrated as follows: — 

Assume 

F(x) = ,/„ + ,/,.,•■' + ./,.)•'• + . . . + ./,,<■'' 
in which ^Z^,, A^, ^t., . . . , do not contain .r, and the exponents 



88 DIFFERENTIAL CALCULUS. 

a,b,c . . . , are written in the order of their magnitude, a being 
the least ; then, by successive differentiation, we have 
F'{x) — aA^x""-^ + hAox'r'^ + . . . -{-2^A^xp-\ 
F'^(x) =a{a — I) A,x"-'' -i-b {h -1) A.,x'-'' + . . . 

Jrp{p-l)A,,xP-^^, 
F"'{x) = a{a — 1) (a — 2) A^x^-'+h{b — 1) (6 — 2) A^x^-^ 
J^..,J^2j(p-l)(p-2)A,x^-\ 



The assumed and all the following equations, being true for 
all values of :r, make x = 0; then, since F{0), F'{0),F''{0) .. ., 
would in general reduce neither to nor to oo, we should have 

A, = F{0), a=l, A, = F^{0), h = 2, 

^2--^^, C-^; A,- ^^^ 

X" X 

F(x)=F{0) + xF'(0)+—F"{0)+--^ F"'(0) + . . ., 

which is identical with the formula of Art. 62. 

70, Taylor's Theorem also admits of the following simple 
demonstration when the function F{x -\- h) can be expanded 
into a series arranged according to the ascending integral 
powers of one of the variables with co-efficients which are 
functions of the other variable only. 

Assume 

Fix + h) =/{x) +/i(x) h' +Mx) 11" + ... +/,(x) h", 

and differentiate with respect to x, and also Avith respect to h; 

then 

dFjx + h) ^d/(x) dMx) ^^^ '^A(^)j.j I ^^ . ^/»(^) ;,, 
dx dx dx dx ' ' ' dx 

dF{x + h) 



dli 



^ aMx)h<^-' + hf,{x)h^-' + , . . -^loMx)hP-\ 



DIFFERENTIAL CO-EFFICIENTS. 89 

But F{x -\- 7i) involves h in precisely the same way that it 
it does x; and, if Ave place x-{-7i — y,WG have (Art. 42) 

dFjxJrh) ^ dF{y) dy ^ dF{y) ^ ^ 

dx dy dx dy 

dFjx + h) ^ dF(y) dy ^ dFjy) ^^^ ^ _ 

d/i dy dh dy 

dF(x + Ji) _dF(x + h) ^ 
dx dh ' 

that is, these differential co-efficients are equal for all values 
of X and 7i, which can only be the case when they are identi- 
cally the same. This requires that 

c=:3, /3(a^)=— --^-^ ...; 
also, by making /z = in the assumed development, we find 
f(x) = F(x); 

whence /. (x) = F'{x), /, (x) = ^^ ...: 

therefore 

F{x-]-h)^F{x)-{-^tF'{x)-{--l^F^^{x)-{- . . . 

12 



SECTION VI. 



EXPANSION OF FUNCTIONS. 



7i, The application of the formulas, demonstrated in the 
preceding section for the expansion of functions, gives rise 
to many important series, some of which we shall now deduce. 
1. If F{x) = {l-^x)^,then 
F'{x) = m{l +x)'"-\ 
F''{x) = m{7n — 1) (1 + xf'-^ 



i^("-i)(x)=m(m — 1) . . . (m— n-|-2)(l -[-ir)™-"+i, 
F^''\x) — m{m — 1) ... (m — 71 + 1) (1 + x)™-"; 
therefore i^(0) =: 1, F'{^) = m, F''{0) =m{m — l)..., 

j/iiu^i) ^0) z= m (m — 1) . . . (m — ?i + 2) ; 
and hence, by Art. 60, 

( 1 + x)"^ = 1 + mx + m -.— ^^' + . . . 

(m — '[)... (m — n-\- 2) „ , 
+ " 1.2.3... (.-1) " 

m(m-l) .. . (m-n4-l)x^ _j_ ^r^« 
1.2.3. .,7^ (i + ^^j . 

When a: is less than unity, the last term in this development 
will diminish as n increases ; and, by making n sufficiently 
great, the series 

1 -{-mx-\-m ■ X' -\-m ^ toq " ^ + • • • 

1. Jt JL. ju, o 



EXPANSION OF FUNCTIONS. 91 

will approximate more and more nearly the true value of 
(1 -{-x)"^ the greater the number of terms taken. 

2. Let F{x) — e^; then 

F'ix) = e" =z F" {x) = F'^' {x)^. . . = F^""-'^ (x) = i^^"^ (x), 
F(0) = 1 := F'^ (0) = F^' (0) z= . . .r=i^(— 1)(0); 

therefore e^ ^ 1 + ^ + ^ + ^3 + • • . 

+ 1.2.3. . .{71- 1)"^" 1.2, . .7^^ * 
Making in this x = 1, we have 

a series that may be used for finding the approximate value 
of e. 

3. Let i^(x) = sin. ic; then 

F^ {x) == cos. X — sin. (x-{-^~\ 

d sin. lx-\-^\ 
f . («l = \-—l = CO.. (« + -) = ,i,,. U + I). 



c^x ^^'\ ' 2 



\ 



7?7r 



Thereforc 7'^(0) = 0, F{0)^\^ F'\^)z=^0, F"\^) = - 1, 



F-^-^\(i)-^^\n^~ln; 



and wo have 



92 DIFFERENTIAL CALCULUS. 



+ 1 .2.3...(»-l )^'°-^~" 

ic'' . / nTt\ 

4. Let i^ (x) = COS. cc ; then 
F'{x) = — sin. x= 003/^ + - V F" {x) — cos.f ic + — - m 

F'" {x) = COS. ^x + 1^ j . . . i^("> (x) z= COS. (^ + ^ 
F{0) = 1, i^^ (0) = 0, i^^^ (0) == - 1; F^'' (0) = ^ 
^(.-i)(0) = cos.^^^7r; 

hence cos. x = 1 — — ^ -}- 



1.2 ' 1.2.3.4 

cc"-i 71 — 1 

H o 7 T^ ^°^* 



1.2.3. . .(/I- 1) 2 

x"' I mi 

' 1.2.3. . .n \ ^2 
By Art. 63, it will be observed that the last terms in Exs. 

2, 3, and 4, diminish as n is increased, and finally vanish when 
n becomes infinite. 

5. Let i^(x) = ^(l +x); then 

^-(^) = (---^^^|-^^^): hence 

i^(0) r= 0, i^^ (j^) = 1, F" (x) = — 1, F''^ {x)z=z2. . . 
i^^»>(0)r=( — l)"-il.2.3. . .(?^- 1); and therefore 

^2 ^3 ^4 / 1 \n-2 

Z(i + x) = x-^+|-^ + ... + ^_-Ai__^«-i 



+ 



2 ' 3 4 ' ' 71-1 

n (1 + (9x)"* 



EXPANSION OF FUNCTIONS. 93 

An examination of tlie last term of this expansion shows, 
that, when x does not exceed unity, this term necessarily de- 
creases as n increases, and vanishes when n becomes infinite. 

And, since the factor ( ^^ p-^- ) under this hypothesis cannot 
exceed unity, the sum of the series, up to the n^^' term in- 
clusive, cannot differ from the true sum by more than - • and 

hence, by increasing n sufficiently, this difference can be made 
as small as we please. 

Changing the sign of x, we have 



n {l — dxf 
6. Let F{x)~i2.i-i.-^x; then 



+ 2X-I xV^'\{i+.W~\)-''- 

F"'{x) 



94 DIFFERENTIAL CALCULUS. 

therefore 

i^« (0) = (v/lTT)'-' ^■'^■^■■■^^''-^) (i _ (_ i)» 

Whence it follows, that, if n is an even number, i^^"^ (0) = ; 
but, if n is uneven, then 

J'"" (0) = (V— 1)"-' l-^-3--^('^-l) X 2 

_(\/-l)"^ 2.3 . . (,i_l) = ± —^1.2.3 . . (n-1). 



V-1 V-1 

Hence we have 

tan -^ x = x ^+^...± r 

x^ (l—dxV^l)-''^(l-\- cV^^)~\ 

The final term in this development is not in a convenient 
form, as it stands, to decide whether the series is converging 
or diverging ; but by referring to Ex. 18, p. 67, making 

a = 1, and observing that there d z=- — tan.~^ x, we have 

i^(«)(^) ^ (_ 1)^-1 llMl-l^iZiZll) sin. ("^ - 71 tan.-'' x): 
therefore 

X X"^ 

tan.~^ X := X — ~ -\- -^ Sz -\- . . . 

o o 

( — 1)'' — sm. ^r- — 7it3in.~^x 

{\ -{- x^y- ^ 

This form of the final term shows, that, if x is less than 
unity, the numerical value of the term may be made as small 
as we please by giving to ?^ a value sufficiently great. 

The above form for F^^^ [x) might have been used for find- 
ing all the differential co-efficients of tan.~^aj as readily as that 
specially deduced for that purpose. 



EXPANSION OF FUNCTIONS. 95 

The following is a more simple process for getting the 
expansion of tan.~^ x : — 

Assume 

t^nr' X = A -Jr Bx -\- Cx^ -j- Dx' + &c. (1), 
and differentiate both members with respect to x ; then 

^ , z=zB-\-2Cx-{- SJDx'' + &c, (2) ; 

JL H — X " 

but by division, or by the Binomial Theorem, 

I -\- X^ 

The second members of (2) and (3) must be identical: hence, 
equating the co-efficients of like powers of x, we have 

' 3' 

and, since the assumed development must be true for all values 
of a;, make ^r = in (1), and we find A ^= : therefore 

tan.~^ o: := x — ^ 1 — \- & — ... 

3^5 7 

7. If y = sin.~^x, assume 

s'm.-'^x = A + Bx-\-Cx''-{- Dx' + . . . (1), 
and differentiate both members ; then 

but, by the Binomial Theorem, we find 

1 . 1 , 1-^ 1.3.5 , 

^7f^^^,-l+.-^'- + 2.I^'+2X6'^ +••• ^^^- 

The co-efficients of iho like powers of x in the second 
members of (2) and (3) must bo 0({ual: hence 

and, by making x — in [\), wo get ^1 = : there tore 
1 x' , 1.3 ic* . 1.3.5 X 



sm. 



^2 3 ^2.4 5 ^2.4.G 7 ^ 



96 



DIFFERENTIAL CALCULUS, 



8. . Let ?/ = e""^'"- "^j and assume 

2/=^o + ^i«^ + -4,a;^ + . . . + ^„a;" + . . . (1). 
Differentiate twice : then 



^y _ 



dx 



= ^1+ 2A,x-\-?>A.x''-\- . . .+nA^x"-^-}- ... (2) 



= 2^2 +"2. 3^3 0:+.. .J^(n-l)nA,x--'' + . . . (3). 
But 



dx 
dx"- 



vr 



= 6 



a sin. "x 



+ 



xae' 



and hence 



(1 -x'')' 



3 1 



:4). 



.^ ^.d~y dy 

(\ — X-) -~^ — x -^ = a-y . 

^ ^ dx'- dx -^ 

Substitute in (4) the values of — , — ^, taken from (2) and 

dx dx^ 

(3), and we have 

2^2+ 2.3^3^; + 3.4^4:^'^ -1 Vin- l)nJ„(r"-- + . . 

— (2^2^^-4-2. 3^ 3^^3+3. 4^,0^4 _^ h(n-l)nJ,ic" + ..) 

-( A^x-^-IA.^x"- ^ZA^x"" ^^A^x' ^ [-^^^«^"+- •) 

aMo + a^A^x + a'-A.^x''' + orA^x 

-{-a-A^x" 



Equating the co-efficients of the same powers of x in the 
two members of this equation, we find 

A--A i-'^'^^A A -""'^^A 

and generally 



(?2 — l)7^ 



• (5). 



EXPANSION OF FUNCTIONS. 97 

If then A^ and A^ be found, formula 5 will give all the 
following co-efficients in terms of these two. 

A^ is what e""''"" ^ becomes when ic = : hence A^^ —1, 

And Ai is what ^ = ^asw-^^: - becomes whence — : 

dx V I — x^ 

hence ^^ = a. 

In formula 5, making n equal to 2, 3, 4, &c., successively, 
we get 

^""1.2' '"" 1.2.3 '■^^4- 1.2.3.4 ••• 

Substituting these values of Ai^, Ai,A2 . . . in the assumed 
development, it becomes 

1.2 "^ 1.2.3 "f" 1.2.3.4 



— - = 1 + ax + f^ + -^f^' X' + '^—'^^ ^' 



a(a^ + l)(a'+8') 

+ — o:t75 ^ + • • • 

By Ex. 2 of this article, wo also have 

\. Z 1. J. o 

Equating the co-efficient of the first power of a in this 
series with tliat of the same power of a in the preceding 
series, we have 

1 x' . 1.3 x' . 1.3.5 x' 



sm. 



I.V, _ 



XziizX -\~ ^- 1 V- „- 4- etc., 

^28 ^2.4 5 ^2.4.G 7 ^ ' 



as in Ex. 7. By equating the co-eilicients of a"-, wo should 
also find 

')2 02 42 02 4.2 (\l 

13 



98 DIFFERENTIAL CALCULUS. 



10. 2/=:Z(l-x + ^^),y=-x + ^ + ?|' + ^-^ 



2 3 

11. y = l{\ + ^m.x), y~x-%^^^ 

12. y = e^-"'-, yz^l+x + ^l--:^--^... 

13. If ?/ = (^ Y~^ — r^ — ] •> s^o^^ fo^ what values of x 

Taylor's Theorem fails to give the developnient. 
It fails for a; = c; 1st term is then infinite. 
It fails for cc =: a; 2d differential co-efficient is then infinite. 



SECTION VII. 

APPLICATION OF SOME OF THE PRECEDING SERIES TO TRIGO- 
NOMETRICAL AND LOGARITHMIC EXPRESSIONS. 

72. Let a and b represent any two real quantities what- 
ever; then a-\-hV— 1 will be the most general symbol for 
quantity, since, by giving to a and b suitable values, it may be 
made to embrace every conceivable quantity, real or imaginary. 

The two expressions, a -\-b V — 1, a — bV — 1, which dif- 
fer only in the signs of their second terms, are said to be conju- 
gate ; and their product, (a -\-bV — I) {a — &V— l) = a^-f-^^ 
is always real and positive. The numerical value of the square 
root of a^ -|- b'^ is the modulus of either of the conjugate ex- 
pressions. Denote this modulus by r; then it may be shown 
that the expression a -\-bV — 1 can be put under the form ■ 

r{cos.d -{-V - I sin. 6). 
Tor let a = r cos. d,b^=r sin. : 

tan.^=: , r-(cos.2 6> + sin.2^)=rr'-=rrt2-f Z>', 
r = Va'' + b\ 
Now, if we suppose the arc of a circle to start from — ~,and 

to increase by continuous degrees to -|- ^^, passing througli 

zero, the tangent will at the same time increase by continuous 
degrees, and pass through all possible values between — oo 
and -|- X) . Among these values of the tangent, there must bo 

one that will satisfy the equation tan. ^^ — ; and the arc an- 

a 

05) 



100 DIFFERENTIAL CALCULUS, 

swering to this tangent will be that whose sine and cosine will 
satisfy the equations a zr^r cos. d^ h:^r sin. d^ and therefore 
render r (cos. d -\- \^ —1 sin. 6) the equivalent of a + ^ V — 1. 
73, Let us resume the series (Art. 71, Exs. 2, 3, 4). 

^"172^+1.2.3.4.0 (^) 



sm. X 



and in (1) write x\ — 1, — x V — 1, for x successively ; then 
+ 1 .o o i > + • • • = cos. a? + V — 1 sin. X, 

1.2.3.4.0 

as is seen by comparing this result with the second members 

of(2)and(3). 

/ . x"' xW-\ , x^ 

Also e-^^-i .=^\-x\/-\——^-\- ^2.3. + i,2.3.4 

^'V^^ , / T • 

— -. ^ ^ . . -f . . • = COS. a; — V — 1 sm. x : 
1.2.3.4.0 

therefore cos. x + V— 1 sin. x = e""^'-^ • • • (4) 
COS. X — V— 1 sin. X = e^-^^^-^ • ■ • (5), 
also COS. ?/ + v^— Isin. y = e^"^-i • • • (6); 

multiplying (4) by (6) 

(cos. X -\- \/ — 1 sin. x) (cos. y + V — 1 sin. y) = e(^+2/)^-i 

= COS. (a; + 2/) + '^"^ 1 ^^^- (^ + y)' 

Effecting the multiplication in the first member, and then 

equating the real part in one member with the real part in the 



TRIGONOMETRICAL EXPRESSIONS, 101 

other, and the imaginary part in the one with the imaginary 

part in the other, we find 

COS. [x-\-y) zzz COS. X COS. y — sin. x sin. y 

sin. [x -\-y) :=! sin. x cos. y -\- sin. y cos. x. 
Again : 

\QO?,.x-\-\ —l&m.xj (cos.y-l-v — Isin.y) (cos.s + V — Isin.sj • • 
_ ^ix+y+z. . )-/-!_ COS. (cc + y + s-l- • •)+ V"^sin. (jc+y + 2 + - • ), 
from which, by making x =iy ^=. z =: • • • , we have 

(cos. X -\- \ — 1 sin. ocj '"" ^ cos. mx -\- ^/ — 1 sin. mx, 
and generally 

(cos. X d= V— 1 sin. x) "^ =z COS. mx i V — 1 sin. mx, 
which is known as De Moivre's Formula. 

Hence the multiplication of expressions of the form of 
COS. X -}- s/ — 1 sin. X, and therefore of all imaginary expres- 
sions, is thus reduced to an addition, and the raising to 
powers to a multiplication. 

7^. Dividing formula (4) of the preceding article by (o) 
of the same, member by member, we have 
e^^~^ _ 2x-^~i — ^'^^' ^ + V— 1 sin, a; _ 1 + V— 1 tan. cc. 
g-zV-TT COS. cc — V— Isin. X 1 — \/^Ttan. .t' 

whence, by taking the Napierian logarithms of both members, 

2xV^^ = l{l-{- V^l tan. a) - I {i - V^^l tan. .r) . 
Expanding the terms in the second member by Ex. 5, Art. 71, 

^ / — - /--r^ , tan.'^a; . tan.^o; t:in.-*.r 

2x V — 1 — V— 1 tan. x + —^^ V— I — ^ j — 




1Q2 DIFFERENTIAL CALCULUS. 

Equating the imaginary parts in the two members of this 
equation, and then dividing through by 2 V — 1, we have 

tan.^a? , tan.^cc tan.'cc , „ „ 
X 1= tan. X — -\ = ^ \- & — <fec., 

a series that may be used for the calculation of rt, and which 
agrees with the formula in Ex. 6, Art. 71. 

75. To find the expansion of cos.^cc in terms of the cosines 
of multiples of x. 

Make e^-^-^ — y; then e'«^^^=: ^z'", e-^-^^i^-, 

From formulas 4, 5, Art. 73, we find 

1 



2 COS. X = e-'"'-' + e-»^-i = y + - 



2 V — Isin.^^e^""-^ _ e--r-'-i — ^ _ -; 

also, from De Moivre's Theorem, we deduce 

1 , 1 

2 COS. mx = y"' + — , 2 V— 1 sin. mx = y^^ — — . 

1 / 1\" 

Because 2cos. xz=v + -, 2^" cos.'^.r =: f ?/ + - ) : 

^ \ y/ 

but 

^ 1\" 77 — 1 

, n-1 1 1 1 

• " + ''172" r^^ + ''r^ + r' 

by combining terms at equal distances from the extremes : 
hence 



TRIGONOMETRICAL EXPRESSIONS. 103 

cos."a; = — --^( COS. nic -|- ?i COS. (n — 2)x 

n — \ \ 

-\-n COS. ('?^ — 4) a:; -|- • • • j {h). 

1.2 / 

Since there are n -\-l terms in series (a), when n is even, the 

number of terms is odd, and the middle term, that is, 

n{n-l){n-2)...(l+2\(^^l 



1.2.3.../^-!^^ 



s2 / 2 

will be independent of y^ and consequently of x; but, when n 
is odd, n + 1 is even, and there is no middle term in series (a), 
and therefore no term independent of x. In the first case, 
there will be within the ( ) in formula (&), besides the term 

that does not depend on ic, - terms, containing as factors the 

A 

first cos. na?, the second cos. {n — 2)x ; and so on to the last, 
which will have cos. 2x for a factor. In the second case, that 
is, when n is odd, there is no term within the ( ) in formula 

(b) that does not involve x; but the ^ terms will then have 
for factors, severally, cos. nx, cos. (n — 2) x . . . , cos. 3x, cos. x. 

Ex. 1. cos.'^o:; =:-— ( cos. 4x -j- 4 COS. 2a:; 4" ^ 

Ex. 2. cos.-'^ic ^ — ( COS. 5^ + 5 cos. o.t + 10 cos. .? 

2^\ ^ ^ 

76, To find the expansion of sin.".r in terms of the sinos 
of multiples of x. 

By formulas 4 and 5 of Art. To, wo have, employing the 
notation of the last article, 

2 V-^l sin. X = e-^^-^ — e"^^"^ = i/ — - : 



104 DIFFERENTIAL CALCULUS. 

1 



therefore 2"(V — 1)'^ sin.^ic = 2"( — 1)' sin.^x z=r ( 2/ — 

if 



— y"- 


-mj'- 


-2 + 71 


n— 1 
1.2 


y-*- 


-&+■■ 


• 






+ { 


-If 


n-l 1 


:! + (■ 


-l)"-'n 


yn-, + ( 


-1)« 


^ 1 


= {y- 


+ (- 


^^"^! 


)-.. 


^2/"-^ 


-(-1)" 


j^-^V 






V- 


i{n — 
1.2 


'^{r- 


-' + (- 


- !)"-■■ 


2/"*V 


& + ■■■ 


(a). 





An examination of this series shows, that, when n is even, 
the second terms within the ( ) are all plus ; and, when n is 
odd, they are all minus. In the first case, the expansion of 
sin."a; will involve only the cosines of multiples of x; and, in 
the second case, it will involve only the sines of these multi- 
ples. 

n 

The factor (—1)^ in the first member will be positive and 
real when n is any one of the alternate even numbers begin- 
ning with ; that is, when n is or 4 or 8 or 12, <fcc. ; and 
negative and real when n is one of the alternate even num- 

n 

bers beginning with 2. In like manner, (—1)"^ will be imagi- 
nary and positive when n is any one of the alternate odd 
numbers beginning with 1 ; and it will be imaginary and nega- 
tive when n is any other odd number. 

Let h represent any positive whole number, zero included ; 
then the different series of values above indicated for n will be 
embraced in the four forms, ih, AJc -\-2] 4^ -|- 1, 4^ -f- 3. 

It would be of no advantage to make formula (a) conform 
to each of these cases by special notation, as it can be easily 
applied, as it now stands, to the examples falling under it. 



TRIGONOMETRICAL EXPRESSIONS. 105 

Ex. 1. Expand sin.^a? in terms of the sines of the mul- 
tiples of iC. 

=z 2\/ — 1 sin. 3x — 6\/— 1 sin. x: 

sin.^ic ;= — — (sin. 3x — 3 sin. ic). 

Ex. 2. Expand sin.'^ic in terms of the cosines of the mul- 
tiples of X. 

2\V^irsm.'x = (y'-i--^ -^(y' + -) + 12 

= 2 COS. 4x — 8 cos. 2x + 12 : 

sin.^o; = ~3 ( cos. 4zx — 4: cos. 2x -\- Gy 

Ex. 3. Expand sin.^;i; in terms of the sines of the mul- 
tiples of x. 

2^(V^l)W-^.-^(2/^-l)-5(y3_l.^ + 10^^-J^ 

— 2\/^^i sin. 5x — 10 V^^ sin. 3x + 20 V^^ sin. x : 

sin.'^x = ^ ( sin. 5x — 5 sin. dx -\-\0 sin. x\. 

Ex. 4. Expand sin.^o; in terms of the cosines of tlie mul- 
tiples of X. 

2«( V- l)«sin.«^ = ^//«+ \^ - C> 6/^+ -) +15 ^/■-'+ i^ _ 20 



/// \ '1/7 \ ir 

— 2 COS. Q>x — 12 COS. l.r -{- 30 cos. 2.r — -JO : 
sin.*''x ■= — ~l COS. G.r — G cos. 4.r -f- 15 cos. 2.r — 1 

77. To find the diirerent n''' roots of unity. 
Let X represent the general value of the n^^^ root o( unity: 
then, hy the definition of the root of a number, .r";:=l, or 

14 



106 DIFFERENTIAL CALCULUS. 

ic" — 1 — 0; and the object of the investigation is to find all of 
the values of a? that will satisfy the equation x"^ — 1 = 0. 
By De Moivre's Theorem, Art. 73, we have 

(cos. y ± \/ — 1 sin. y)^= cos. my dz V — 1 sin. my; 
an equation which holds, whether m is entire or fractional, 
positive or negative. Now, if Ic be any whole number, 2k7t 
will be an exact number of circumferences to the radius unity, 
and 

COS. {y -\- 2k7t) = COS. y, sin. (y -[- 2k7t) =r sin. y : 

therefore (cos. ^ =±z V— 1 sin. ^)"* 

— ("cos. {y + 2k7t) ± V^^ sin. {y + 2k7t)y. 

1 1 

Make m r= - ; then (cos. ^ =1= V— 1 sin. y)" 

1 

= ('cos. {y + 2k7r) ± V — 1 sin. {y + 2k7t)y 

y -^2k7t , — - . y -^2kTt 

= COS. =b V — 1 sm. . 

n n 

In this last equation, make y zzzO: whence, as cos. 2k7t ^= 1, 

and sin. 2k7t = 0, we have 

(l)'^ = COS. d= V — 1 sm. . 

n n 

1 
But, from the equation cc"— IrziO, we get x^^(\y: hence 

we conclude that the different values of x, or the roots of the 

equation a:;" — 1 = 0, are the values that may be assumed by 

2kTt . , — - , 2kTt 

cos. ± V — 1 sm. 

n n 

by assigning different values to k. Since k may be any 

whole number, take for it successively 0, 1, 2, <fec. ; then, 

1 ' 

when Zj = 0, 1^ = cos. d= V— 1 sin. = 1, 

1 27t , / . 27t 

when A; = 1, 1" := cos. ■ — =r: v — 1 sin. — , 



TRIGONOMETRICAL EXPRESSIONS. 107 

L 4:7? . 4:7t 

when ^ := 2, I'^^cos. — =t>\/— Isin. — , 

. ; 

and so on, continuing the substitutions for h until the 

2^^ , . . 

arc reaches such a value as to cause the expression 

n 

2k7t , / — - . Ikrt ^ ^ • 1 1 -, 

cos. =b V — 1 sm. to reproduce the roots it has already 

n n 

given. When n is an even number, this will be the case 
for ^ = ^ ; for, 

-, n ^ ^^ n — 2 , / — -- . n — 2 

if A3 = - — 1, l"=cos. 7? =t V — Ism. 7?; 

2 n n 

n I , 

if A; =: -, 1" =: COS. Tt =t v — 1 sin. n = — 1 ; 

A 

, n ^ i 71 + 2 , / . 72 + 2 

if A: = -4-1 1" =z COS. ^ TT =b V — 1 sm. n : 

z n n 

n—2 , / — - . n — 2 

but COS. 7? dr V — 1 sm. 7? 

n n 

71 + 2 . n + 2 

= COS. 7? =p V — 1 sin. Tt : 

n n 

therefore the two roots corresponding to A' ^ - -f- 1 are the 
same as those corresponding to A: = - — 1. So, also, those ob- 
tained by substituting ,^ + 2 for h are equal to those obtained 

71 

by substituting ^^ — 2 for A;, and so on: whence all substitn- 

tions for h after the value ^ would merely reproduce the rocUs 
already found. 

Again: when n is an odd number, the substitutions for k 

must be continued until k := » ; for, 

2 



108 DIFFERENTIAL CALCULUS. 

11 Ic— — ^r— , I'^^cos. TTzizV — Ism. 7t: 

if Tc— -^ ^1, l« = cos.-^^I^7?=h V— Ism. — -^— -7t; 
, , ?z — 1 , ^ / — - .n — 1 n -\-l 

but COS.- — 7r±V — Ism. ;r = cos. — ^ — tt 

^ n n 

/ — . . ?i + l 
=FV — Ism. — — tt; 

n 

hence the substitutions of — ^ — and - "^ for k give the 

same roots. So, also, it may be shown that the substitutions of 

^ 3 72,-1-3 

— ^— and — ^ — - for k would give the same roots. Therefore 

we should merely reproduce the roots already found, if we 

n-1 



substituted values for k greater than k 



2 



When n is even, ^ = and k ^=^ - give, the first the root 

-|- 1, and the second the root — 1 ; and the intermediate 
values of k give each two roots. When n is odd, A; = 

72, \ 

gives the root 1 ; and all the other values of k, up to — - — 

inclusively, give each two roots. In either case, the expres- 

o7,^ iyUjf 

sion COS. — ~ d= V— 1 sin. can assume n different values, 

n n 

and no more. Hence it follows that the equation x^' — 1 = 

has n dijBferent roots, and can have no more. 

By the aid of the foregoing principles, the roots of tlie 
equation x^^ — 1 i= may be expressed under the form of ex- 
ponentials. 

Since, by Eqs. 4, 5, Art. 73, we have 

cos. a:; =t V — 1 sm. x^^ e - j 



TRIGONOMETRICAL EXPRESSIONS. 109 

the successive values taken by the expression 

2kn: / 7 . 2k7t 

COS. — ■ =h V — 1 sm. 

n n 

may be represented in order by 

when n is even, and by 

e , 6^ '^ ,6 .. ., e « S 

when 7^ is odd ; the first term in each series of roots being 
unity, but the last term in the first series is minus 1, since it 
is equal to cos. tz: ± V— 1 sin. 7t =: — 1. Both series of roots 
are the terms of a geometrical progression, the first term 



of which is e =1, and of which the ratio is e 



/ — — *^— 1 

•J 

Ex. 1. What are the three cube-roots of unity? 

They are the roots of the equation a;^ — 1 = 0. 

Here tz = 3, and the proper values for h in the expres- 

2kTt , OZ-TT 

sion COS. -— d- V _ 1 sm. — -^ are O-and 1: hence the first 
n ^i 

gives 

(if = COS. i V=l sin. = e±'^~^ = 1. 
The second gives 

(l)^z=cos., dzV- lsin.';^ = 6^ 3-'-\ 

Ex. 2. Find the roots of x'' -1=0. 
Hero n = 6, and the proper values for h are 0, 1,2, 3. 
(1)^ = cos. ± V=T sin. = e ±«^^^ i. 

It 



(l)c = COS. ,, -i^ V— 1 sin. o — c^ 



-3-^-V 



) O 



(l)i = cos. ; iV- lsin.-*;;^ = e*3 ^-i. 
o o 

(l)^ = cos.;r±>/- ls\n.n = e^''^'~' = - \, 



110 DIFFERENTIAL CALCULUS. 

For each root of the equation a;'' — 1 =r 0, there is a binomi- 
al factor of the first degree with respect to x in the first 
member of the equation. Since A; = gives but one root, 
unity, there will be but one corresponding factor x — 1 : ^ = 1 
gives two roots, and the corresponding factors are 

/ 27t , / — . . 2n\ I In /-— - . 2;t\ 

X — [ COS. h V — 1 sm. — Vx ~[ cos. v — 1 sm. — , 

\ n nj \ n nj 

which by multiplication will produce the quadratic factor 

In 

x"^ — 2xcos. \-l. 

n 

In like manner, each pair of simple factors may be reduced to 
a quadratic factor. \in is even, the last factor \& x -\- 1^ which 
may be combined with the first factor x — 1, producing the 
quadratic factor x'^ — 1. Hence, when n is even, we have 

x^-l z=l{x'' -l)[x'' — 2x COS. — + 1^ (x" — 2x COS. — + 1 

X- — 2X COS. Tt + 1 

\ n 

and, when n is odd, 

x"" — l — [x — Vjlx"- — 2x COS. — + 1 j lx'^ — 2x cos. — + 1 

o n — \ 

Ix COS. n 

n 

Ex.1. {x^—\)={x-l)[x''-2xGo^. 
Ex. 2. {x^ - 1) 

^ {x' - 1) fx'' - 2x COS. I + 1^ U'' - 2x COS. ~ + 1 

78. The solution of the equation ic'^ + 1 r=z 0, and the reso- 
lution of its first member into factors. 
Eesume the equation 

, I ?/ + 2kn / — T • y + 2k7t 

(cos. y dz V- 1 sin. y)« = cos. ^—^ =b V - 1 sm. ^-J- 



TRIGONOMETRICAL EXPRESSIONS. Ill 

of Art. 77, and make yz=z7t; then, since cos. tt^— 1, and 
sin. 7t =: 0, this equation becomes 

^ ^,i 2h^l / — - . 2/^+1 
(— I)" — COS. — Ttzb V — 1 sm. ^ — TC. 

1 
But, fromi:c" -[- 1 — 0, we have £c = (— 1 )«; hence the roots of 

the equation a?'* -|- 1 =: are the values of which the expres- 

2^ + 1 _u./ — 1 • 2^^4-1 „, T ., . 

sion cos. ' — ;t =1= V — 1 sm. — — ' — 7t will admit lor ad- 

n n 

missible values of h. But h may be any whole number in- 
cluding zero. Therefore take for h successively the values 
0, 1 , 2 ... ; then, 

for^=:0, (—lY— COS. -zb V— 1 sin.-: 

n n' 

fory^=l, (— l)"-=cos. — db V— 1 sin. — ; 



for A; = -— 1, ( — 1) n ~ COS. 7t dz V — 1 sm. n. 

^ n n 

When n is even, substitutions for h greater than ^ — 1 will 
only reproduce preceding values for (— 1)'S for, if k — -^ 

Li 

then (- 1)'' = cosJn +"\ ± V^T sin. /« -f- " 

which is the same pair of roots as that given by the substitu- 
tion of ^ — 1 for h. In like manner, it may bo shown, that, if 

^^ = 2 + ^^ the pair of roots would bo the same as that for 
fc = - — 2 ; and so on. 



112 DIFFERENTIAL CALCULUS. 



When n is odd, the substitutions for k must be continued 
until h = — - — ; for, 

\i h = — :,— , (— 1)"= COS. TT-t- V— Ism. n ; 

n-1 - 

[^ h z= — - — , ( — 1 ) " = COS. :7r i y' 1 sin. tt = — 1. 

A 

Now, for the next value of k, that is, k = — [- 1 — , 

(- 1)^= COS. !^±^ « ± V^ si». !i±^ „ 

n n 

n — 2 / — zr . n — 2 

— COS. TT =F V — 1 sm. 7t : 

n n 

and therefore this substitution for k gives the same pair of 

roots as is given for ^ = — —, and the higher values of k 

merely cause preceding pairs of roots to recur. Hence, 

whether n be even or odd, there will be n, and onlj n, differ- 

2 
ent values for (— 1)" ; and the equation x^ -\-lz:z{) has n, and 

only Uj different roots. These roots can be put under the form 

of exponentials, as in the case of the roots of x" — 1 = 0. 

Ex. 1. What are the roots of ^^ -|- 1 == ? 

Here n = 4 ; and the formula 

. .1 2^+1 . ,/— 1 . 2A^+l 

(— 1)« = cos. — It =b V — 1 sm — Tt 

n n 

gives, for k=.()y (— 1)^= cos. jdr V— 1 sin,^; 

for A; =: 1, ( — 1 ) " = cos. -^ =b V-- 1 sin. -y-. 

4 4 



TRIGONOMETRICAL EXPRESSIONS. 113 

Ex. 2. What are the roots of ic^ + 1 = ? 
Here n = 5 ; and the formula gives, 

for^ = 0, (-1)" z=cos.^±\/^^sin.f; 

5 

for k = l, {~iy z= COS. -.- ± V- 1 sm. -5-? 

1 

for k — 2, {—lf = cos.7r^V—lsin.7Z= — 1. 

For each root of the equation x'^ -{- 1 =^ 0, there is a corre- 
sponding binomial factor of the first degree with respect to x 
in the first member of the equation. 

When n is even, all the roots enter the equation by conjugate 
pairs, and the factors of the first member, answering to the 
simple roots of each pair, may be compounded into a rational 
quadratic factor, and we should have 

x"' -\- 1 ^ I x"^ — 2x COS. — \-l) Ix''^ — 2x cos. — -|- 1 
\ n / \ n 

o n — 1 , . 

— ZX COS. 7? -f- 1 

n 

When n is odd, there will be rational quadratic factors for 

^ 3 

A; = 0, A: = 1 . . . , up to h = inclusively ; but, for h == 

ill 

n — 1 
~— , there is only the simple flictor ^ + 1 ; so that, in this 

case, we should have 

X'" + 1 == fx'' — 2x COS. - + 1) ('V' - '2.t' COS. ^"^ + 1 

x' - 2x COS. ^-^:~— ;r -f 1 V.r -I- n. 

The solution of the equations x" — a = 0, x" -j- a ^- 0, and 
the resolution of their first moinbors into simple and quadratic 

15 



114 DIFFERENTIAL CALCULUS. 

factors, may be at once effected by the formulas in this and the 
preceding articles : for these equations give respectively 

11^ 11 

x — a^{lY, xz=:{aY { — ly, 

1 . 
in both of which a 'i is the numerical value of the 7i* root of a; 

1 
andthisj multiplied by the different values of (1)", will give 

the roots of x^ — a == ; and, multiplied by the values of 

(— l)'^j will give the roots of x" -|- a = 0. 

79. The determination of a general expression for the log- 
arithm of a number positive or negative. 

In any system of logarithms, the logarithm of 1 is 0, and the 
logarithm of is — oo if the base is greater than unity, and 
-[- oo if the base is less than unity ; while the logarithm of oo 
is -|- oo or — oo, according as the base is greater or less than 
unity. It thus appears, that, whatever be the system, all pos- 
sible positive numbers between and oo will embrace for their 
logarithms all possible numbers between — oo and -[- oo. The 
logarithms of negative numbers, if they admit of expres- 
sion, must therefore fall in the class of imaginary quantities. 

In the equation 

COS. X + V^^ sin. X = e^^^ (Eq. 4, Art. 73), 
write X -|- 2k7Z for x, h being any whole number ; then 

COS. {x + 2h7t) + V^^ sin. (x -f 2k7t) — e^^ + 2^-^) ^^. 
For x=iO, this gives 1 = gS^TrV^i . 

for X=:7t, this gives — 1 := g(2/l' + l)7r^-l ^ 

Taking the Napierian logarithms of both members of these 
equations, we have 

l{l)=:2k7tV^^, l{-l) = {2h+ l)7r\/^T. 



LOGARITHMIC EXPRESSIONS. 115 

These are the general expressions for the Napierian logarithms 
of 1 and — 1 : and, since h may be any whole number, it fol- 
lows that both + 1 and — 1 have an infinite number of log- 
arithms ; but all of them, except that of + 1, corresponding to 
A; = 0, will be imaginary. 

From this it may be shown, that any positive or negative 
number, in whatever system, has an indefinite number of 
logarithms. 

For, first, suppose y to be any positive number, and x its 
arithmetical logarithm taken in the Napierian system; then 
?/ = e^ 1= e^ X 1 = e^ X e2/-^^-i = g^+2/J--^-i . 
ly ~x-[-2'kTt\^ — I, 
which is the general Napierian logarithm of y, and will ad- 
mit of an unlimited number of values. Denoting the arith- 
metical logarithm by l{y), we have 

'^y = Hy) + 2A^^ V^l . . . (m). 

Again: let Ly denote the general logarithm of y, taken in 
the system of which a is the base, L (y) denoting the arith- 
metical logarithm ; then, since we pass from Napierian to any 
other logarithms by multiplying the former by the modulus of 

the system to which we pass, multiply Eq. m by --, the 

LCI 

modulus of the system characterized by L, which gives 

or, 

2k.7 v/- 1 
Ly = L{y)+ ^^^ . .. (n). 

From Eqs. m, n, wo conclude that the arithmetical logarithm 
of a positive number taken in any system is the value of the 
general logarithm corresponding to k = 0. 



116 DIFFERENTIAL CALCULUS. 

Now, suppose y to be negative ; then —y= — lxy, and 
— lX.y = e^X— l==e"^X e^2^-+i)^"^-i = g.r+(2^-+i)^-^~i : 

also X(— y) =: — I-^ n__Z (g). 

Eqs. p, q, are the general expressions of the logarithms of a 
negative number, and show that such a number has an unlim- 
ited number of logarithms, all of which are imaginary. 

From the equation l{ — l) = (2h-\-l)7e\/—l,we get 

(2^ + l)V-l 
This and the preceding remarkable results developed in this 
section must be interpreted with reference to the symbols 
and the character of the quantities with which we are dealing. 
It must be remembered that e and 7t are the representatives 
of arithmetical series, and that the formulas have meaning, 
and can be regarded as expressing true relations, only when 
the rules for combining imaginary quantities with each other 
and with real quantities are strictly observed. 



SECTION VIII. 

DIFFERENTIATION OF EXPLICIT FUNCTIONS OF TWO OR MORE IN- 
DEPENDENT VARIABLES, OF FUNCTIONS OF FUNCTIONS, AND OF 
IMPLICIT FUNCTIONS OF SEVERAL VARIABLES. 

80. When several variables are involved in an equation, 
any one of them may be selected as the function or dependent 
variable ; the others being regarded as independent. If the 
value of the function is directly expressed in terms of the va- 
riables, we have an explicit function of several independent 
variables ; but, when the function and the variables are in- 
volved in an unresolved equation, we have an implicit function. 

Let u = F{Xj y) be an explicit function of the two independ- 
ent variables, x^ y^ and give to these variables the increments, 
AX, A?/, whereby u receives the increment ^u expressed by the 
equation 

AW = F{x + ^x, y^Mj)- F{x, y) = F{x + A.r, //) - F[x, y) 
-^F{x-^Ax,y^Ay)-F[x^i.r,y). . .(a). 
The partial derivative, or difibrential co-efficient, of n function 
with respect to one of the variables involved in the I'unction, 
is that which comes from attributing an increment to that va- 
riable alone. The partial derivative, or ditferential co-onicient, 
of 'u ^= F{x, y), taken with respect to x, is denoted by F'^.[x, //\ 

^^f' I Tl 7 7'/ N i^(< 1 ,, • 1 1-,. 

or- . In hke manner, /^,.(.r,v/), or , , dcMiotes (ho ivirtial dit- 
dx . V - . / ^^^^ ■ 

ferential co-ellicient taken with respect to // ; and Fj,\x, y^yOV 



118 DIFFERENTIAL CALCULUS. 

is the partial differential co-efficient taken with respect 

dxdy 

to X of the partial differential co-efficient taken with respect 

to y. 

Now, if ri, r.,; r.^^ are quantities which vanish with a a?, a^, 

then, by Art. 15, we have the following: — 

F{x + AX, y) — F{x, y) = F'^{x, y)^x + r^ t^x, 

F{x-^^x,y + ^y)-F{x^^x,y)z:zFy{x^^x,y)^y^r^^y, 

Fl{x + AX, tj) - F;^{x,y) = F;'^{x,y) AX i- r,Ax ; 

from which last we get 

Fy{x + AX, y) - F'y{x, y) ^F'^^yi^x, y) Ax + r^Ax. 

By substituting these values in Eq. a, it becomes 

Au —F'^{x, y) AX + Fy{x, y) Ay + r^ ax + r,Ay 

+ K'y{^,y)^^^y + r,AxAy; 

or, 

da du . 

AU ^ ^- AX -{- -^ Ay-\-riAx -j-r^Ay 
ax ^y 

The increment az^ of a function of two independent varia- 
bles is, therefore, like that of a function of a single variable, 

composed of two parts ; the one, — ax -\- -=-Ay, of the first 

(XX cty 

degree with respect to the increments ax, Ay, and in which 
the co-efficients of these increments do not vanish with the 
increments. The other part is made up of terms which are 
either of a higher degree than the first with respect to ax, Ay, 
or they are terms in which the co-efficients Ti, To, r^, of the first 
powers of AX, Ay, vanish with these increments. 

From what precedes, we pass by what seems to be a natu- 
ral extension of our definition. Art. 16, of the differential of a 



DIFFERENTIATION OF EXPLICIT FUNCTIONS. 119 

function of a single variable, to that of a function of two varia- 
bles. If we write du^ dx^ dy^ for Lu^ lx^ Ly^ respectively, in 
Eq. 5, neglecting at the same time all the terms in the second 
member after the second term, we have 

du ^ du ^ 

Here du in the first member denotes the total differential of 

u. and is different from the du m -^i i— In this, as in former 

ax ay 

n i-nc^ . . du du . _ _ , ,. . 

cases of differentiation, -^7 -r-, are to be regarded as the limits 

of the ratios of the increments of the variables to the corre- 
sponding increments of the function; the distinction being, that 
now" in each of these ratios the increment of the function is 
partial, and refers to the variable whose increment is the 

du du 
denominator of the ratio. We must treat -r-, -^7 as wholes, 

and not as fractions having du for the numerators, and dx^ dy^ 
for the denominators. It is true that du^ dx, dy, in these differ- 
ential co-efficients, may be regarded as quantities rather than 
as the traces of quantities which have vanished, by assigning 
them such relative values, generally infinitely small, that their 
ratio shall always be equal to the differential co-efficients. In 

this case, y- dx would reduce to du; but this is the ]iartial 

differential oi u taken with respect to .r, and should be written 

du 
dj.u. So likewise . dy should bo written d„u. To indicate 

du du . 1 T V. .1 r.' . 

that -.) T } are i)a,rtial diQerential co-euicients, they are soiuo- 

times encwsed lu ( ) ; thus, , , , . 

\dx/ \dy/ 

From Vai. c, wo conclude that the total differential of a fuiu'- 



120 DIFFERENTIAL CALCULUS. 

tion of two independent variables is the sum of the partial 
differentials taken with respect to each of the variables sep- 
arately. 

SI, To find the differential of z^ = F{x,y^ z), a function of 
the three independent variables x, y, z, denote as before, by 
^17 ^2J ^3 • • • 7 quantities that vanish with ax, Ay, az; then 

Au = F(x + Ax, y + Ay, z ^ az) - F{x,y, z) 

— F{x + AX, y, z) - F{x, y, z) + F{x + Aa:, y -\- Ay, z) 

—F{x-^Ax,y,z)-\-F{x + Ax, y ^ Ay, z -^ az) 

-F{x + Ax, yJ^Ay,z).,. (d). 
But, Art. 16, 

F{x + AX, y, z) — F{x, y, z) = F^ (x, y, z) ax + r^ ax, 
F{x -^Ax,y-i- Ay, z) — F{x + ax, y, z) 

F(x-\-Ax,y-\-Ay,z-\-Az)^ _, 

^ "T, '-^7 ^'7 [[=F:{x+Ax,y+Ay,z)Az-\-T,Az. 
— F{x-\-Ax,y-^Ay,z)) 

Also, from same article, 

F'y{x + AX, y, z) - Fy{x, y, z) = F['y{x, y, z) ax + r^Ax; 
and therefore 

Fy{x + AX, y, z) = F'y{x, y, z) + F^.y{x, y, z) ax + r^ ax. 
So, likewise, 
K (^ + Ar^-, y-\-Ay,z) = F^ {x -^ax, y, z) 

-\-Fy,{x-\-Ax,y, z)Ay + r^Ay, 
and 

F',{x + Ax, y, z) — F[{x, y, z) -^F'^,{x, y, z)Ax-\-r^Ax. 

Making these substitutions in Eq. d, and denoting the co- 
efficients of the terms containing the products of Ax, Ay, az, 
by each other, by m^, m^, m^, we have 

Au = Fl.{x,y, z)Ax-^F'y{x,y, z) Ay -\- F[{x,y, z) az 
-\-rYAx-\-Try_i^y -\rr-iAz ^m^Ax Ay -^m^Ax Az-^m-^Ay az; 



DIFFERENTIATION OF EXPLICIT FUNCTIONS. 121 

du , du du , 

^''^ ^^ "^ dx^"^ ^ dv^^ '^ ~dz ^^ ^ "''^'^ ^ '''^^ '^ '^'^ '^ 

-{- rrii A X A y -\- m2 ^ oc A z -\- m ^ A y A z. 

From this, by the same considerations that led ns to the 
expression for the total differential of a function of two inde- 
pendent variables, we conclude that 

, du J , du J , dio J 
du = .=- dx -\- ,- a?/ + ,- dz, 
dx ^ dy -^ ^ dz ' 

which may be written 

dvb = dji -\- dyU -\- d^u. 

The course to be followed for a function of four or a greater 
number of independent variables, and the results at which we 
should arrive, are obvious. The total differential of a function 
of any number of independent variables is therefore equal to 
the sum of the partial differentials of the function taken with 
respect to each of the variables separately. 

S2, In Art. 42, a rule was given for the differentiation of a 
function of an explicit function of a single variable. It is now 
proposed to treat this subject more generally. 

Let u = F{ij, z) be a function of the variables y, z, which 
are themselves functions of a tliird variable x, and given by 
the equations y =^ cp {x), z = ip (x). If x be increased by a j:, 
u, y, and z will take corresponding increments, which denote 
by All, Ay, Az ; then 

Auz= F{y + A //,:>; + A 2) — F{y, z) =r F[y + Ay, z) 

-F{y, z) + F{y + a//, z + a .) - F ^y + a//, ::). 
Dividing through by Aa-, and in the second member multiplying 
and dividing the first two terms by a//, and tlie second two by a,;. 
Aio _ F(y -f- Ay, z) — F[y, ::) a // 
Ax~ Ay AX 

F{y^Ay, z-}-Az) — F[y + Ay,z) az 



AZ AX 



IG 



122 DIFFERENTIAL CALCULUS. 

Passing to the limit by making aoj =: 0^ and remembering 
that ^y and as vanish with A a:, the first member becomes 

— : the first term of the second member becomes — — =^ . To 
ax ay ax 

see clearly what the second term of the second member be- 
comes, suppose, first, that ^y vanishes ; then this term reduces 

to 

F{y,z^Az)-F{y,z) as. 

A2 Aic' 

and it is evident that, now, the factor — ^ — — — ^-^ V^^ 

' ' AS 

is the ratio of the increment as to the corresponding incre- 
ment of the function : hence, at the limit, this factor becomes 

-- , and the second term -.= — =- ; and therefore Ave have 
dz dz ax 

du du dy du dz 

dx dy dx dz dx^ 

and 

du y 7 du T , du -. 
--- dx := du ^=^ —- dy -\- —- dz. 
dx dy dz 

In general, if to =i^(?/, z,u^v . . .), y, z^ u^ v . . , being all 
functions of the same variable x, we should have 

dio dw dy dw dz div du . . 

dx dy dx dz dx du dx 

-. dw 7 dw -, ^ diD -, . ,j. 

dw z= cZ?/ -f -— dz J^ -du+ . • • (b). 
dy dz die 

-TT diu dw dio -, dw -, ,. ,- i -i-rp 

Here ~^, -^r- - ■ - j -^di/, -^-dz, are the partial dilieren- 
dy' dz dy ^' dz ' 

tial co-efficients and partial differentials of the function w; 

while -y- and div in the first members of these equations 
dx 

are the total differential co- efficient and total difi'erential 



FUNCTIONS OF FUNCTIONS. 123 

of the function : hence we may enunciate the following the- 
orem ; viz., the differential co-efScient of a function of any 
number of variables, all of which are functions of the same in- 
dependent variable, is the algebraic sum of the results obtained 
by multiplying the partial differential co-efficient of the func- 
tion taken with respect to each dependent variable by the dif 
ferential co-efficient of such variable taken with respect to the 
independent variable. This is the meaning of Eq. a; and 
Eq. h admits of a like interpretation. 

If in the function, u:=^ Fi^y^ z), we suppose, for a particular 
case, that y and z in terms of x are given by the equations 

y=i f(x), z ^^x ; then dz = dx, — - =: 1, — - i= — - ; and the sec- 
^ -^ ^ ^' ' ' dx ' dz dx' 

ond term in the second member of the equation, 

du du dy du dz 

dx dy dx dz dx' 

du 
would reduce to -^, which is the partial differential co-efficient 

of t^- = F{ij^ z) zzz F{y, x) taken with respect to x. This 

dUf 
must be in some way distinguished from -j- in the first mem- 

cix 

ber of the equation, which is the total differential co-efficient of 
the function. This is usually done, in cases where the two 
kinds of differential co-efficients are likely to be confounded, 
by enclosing the partial differential co-efficients in a paren- 
thesis. Thus the above equation should then be written 



dit _ /df(\ dy /^^'A 
dx^y^dy/dx^^dA-/ 



83, It may happen that some of the subordinate functions 
are themselves functions of the others, and thus conipHcato 
the example ; but the principle just demonst rated is easily 
extended to such cases. For example : — 



124 DIFFERENTIAL CALCULUS 

let u = F {y, z, v, x), v =f{y, 2, x), 

y = (p (x), z=zyj (x); 

from which, by making the proper substitutions, u could be 
made an explicit function of x, and thus the differential co-effi- 
cient of w with respect to x be found. But this result may be 
reached without making these substitutions. 

Differentiating each of these equations Avith respect to x, 
we have 

du /du\ dy /du\ dz /du\ dv ^ /du" 



dx \dyjdx \dzjdx \dvjdx \dxj^ 

dv __ /dv\ dy /dv\ dz /dv\ 
dx ~ \dy) dx \dzj dx \dxj^ 



dx 



in which we distinguish partial from total differential co-effi- 
cients by enclosing the former in parentheses. By substituting 

dv dz/ 
in the first of these differential equations the values of -^, -^, 

dz 

-^, derived from the others, we get, finally, 



du /du\ , , . , /du\ , . , 

4i=W'' w + (s)'•<'' 




*),..,^-(S),.-<.)+(|) 



Ex. ^^ — y2^g3_|_22^^ 




y =r cos. x^ z ^ e 

dy-^y^.^ ^ dz 

dii . dz 

~ z^ — sm. X, -^ 
dx ' dx 



~ = 2y + ^', -j:- = Sz' + 2zy, 



IMPLICIT FUNCTIONS OF VARIABLES. 125 

therefore ^ =z — (2y + 2^) sin. x + (Sa^ 4- 22y) e^ 
dec 

= — (2 COS. aj -|- e^^) sin. a:; + (3e^^-|-2e^cos. ic) e^ 
z= 3e^-^ — e^^ (sin. x — 1 cos. x) — sin. 2x ; 
a result identical with that obtained by first substituting in 
w the values of y and 2, and differentiating the explicit function, 
u = cos.^ x -{- e^^ -\- e^^ cos. x. 
84, When the relation between the variables is expressed 
by an unresolved equation, any one of the variables may be 
assumed as a function of the others regarded as independent. 
It is often inconvenient, or even impossible, to solve the equa- 
tion with reference to the variable taken as the function, and 
thus convert it into an explicit function to which preceding- 
rules for differentiation are applicable ; and hence the necessity 
for investigating special methods for the differentiation of this 
class of functions. 

Consider, first, a function of a single variable, which, in its 
most general form, may be written u = ^{jo, ?/) = 0. Either 
y may be taken as a function of x, or a; as a function of y. It 
generalizes our result to leave the selection of the independent 
variable undetermined. Let ax, Ay, be the simultaneous incre- 
ments of X and y. The increased variables x -\- ax, y-{-Ay, 
are subject to the law of the function F{x,y) = 0, and hence 
must satisfy the equation, 

F{x-\-Ax, ?/ + A?/)=:0: 
therefore 

Au— F{x + Ax, y -\- Ay) — F[x, y) = 0. 

Treating F{x -}- ax, y -{- Ay) — F[x, y) as was done in tlio 
case of a function of two independent variables in the last 
article, we have 
AU = 

die ^ . die . , , d-u. 

= ,7^^'^'+ .„-VV + n-^-^- + '\.A//+ -AxAy^r.AxAy.. ^a)\ 
ax ay axay 

''*u ^'ij ^'a; being quantities that vanish with A.r, Ay. 



126 DIFFERENTIAL CALCULUS. 

Now, by wliichever of the increments we divide tliroTigh, 
and then pass to the limit, by making that increment zero, it is 
manifest, that since, from the mutual dependence of x and y, 
Lx and Ay become zero together, all the terms in the second 
member of the above equation will vanish except the first 
two. 

Dividing through by Lx, and passing to the limit, we have 

du du Ly du du dy 

dx dy ' Lx dx dy dx 

du 

, dy dx 

whence -^ = _ — - — . 

dx du 

dy 
Dividing through by Ay, and passing to the limit, we get 

du 

du dx du ^ ^ . dx dy 

dx dy~^ dy~~ ' ' ' dy~ du 

dx 
In Eq. a, writing du, dx, dy, for ^u, t\x, Ay, and omitting 
all the terms in the second member after the first two, it be- 
comes 

, du T , du ^ ^ 

du ■=^ ~- dx -\- -- a?/ = 0. 
dx dy ^ 

85, li u^=i F[x,y^% . . .) = be a function of any number 
of variables, one among them may be taken as a function of 
all the others regarded as independent. Were the equation 
solved with reference to the variable selected as dependent, 
we should then have to deal with an explicit function of several 
independent variables, — a function which has no total differen- 
tial co-efficients, such as there are in the case of explicit func- 
tions of a single variable ; and we are, therefore, concerned only 



IMPLICIT FUNCTIONS OF VARIABLES. 127 

with the total differentials of the function, and with its partial 
differential co-efficients of the different orders. 

Suppose z to be the dependent variable, and that the value 
of s, in terms of the other variables, is z z=Lf{x^y. . .) : then 

uz:^F(x,y,f{x,y. ..)•.•) = 0; 

and, considered with reference to x alone, i^ is a function of x, 
and of a function of a function of x. But, by the law of the 
function F[x,y^z . . .), u must be zero for all values of the inde- 
pendent variables : hence its partial differential co-efficients, 
taken with respect to these variables, must be zero. 



Denote ^J \-^) the partial differential co- efficient of u 
taken with respect to x, and, through x, with respect to z; and, 
by -^, -^, the partial differential co-efficient taken with re- 
spect to X and z separately : then, by Art. 82, 

du\ die du dz _ 

dx) dx dz dx 
Similarly, by adopting a like notation with reference to 
y,Sjt . . ., we have 

/du\ ___du dibd% _^ J 
\dy/ ~~' dy~^ dzdy~ 
/du\ dio du dz 



\ds J ds ~^ dz ds 



Eqs. a,h, c . . ., will give the partial differential co-efficients 

of z with respect to the variables severally. Thus, from {a), 

we have 

du du 

dz dx dz du 

,— , , and, from (h), , =: _ / - • • • 

djc du' ^ dy du 

dz dz 



128 DIFFERENTIAL CALCULUS. 

Multiplying Eqs. a,h,c... through by dx, dy, ds . , . respec- 

tively, and adding the results, observing that 

du dz -. , du dz ^ , du dz j , du ■, 

-— dx-\- ^ — ^-au 4- ^ — =- OS +.•.:= ^— a2, 
dz dx dz dy dz ds dz 

we have 

du T , da T ^ du T . du T ^ . , , 

d.'^"+ d^'^^+ d^^^ + rfT*+ • • • =0 • • ■ (-)• 

From Eq. m, we may find the total differential of any 
one of the variables regarded as a function of all the others; 
du . , du y , du ^ , , 

*^"' rf.,- T/'J + di'^'-^d^'^' + --- 



du 
dx 

Ex. 1. u = ay + hV - a'o' = 0, 

therefore a^;f$ -\-b^-x = .■ /^= -^^ . 

dx dx a^y 

From the given equation, we get y — - Va^ — x'^, an ex- 

a 

plicit function of y; and, by differentiation, we obtain directly 

dy hx h^x 



Ex. 2. 



dx 


a*^ o? — x'^ 


a'^y 


U — 


y^ ^ x"^ — Zaxy -- 


= 0, 


du _ 
dx 


:3.'-8a,,| = 


3z/^-- 


dy _ 


x"^ — ay _ ay 


-x'^ 



3ax, 



dx y'^ — ax y- — ax 

a result that it would be difficult to verify, as was done in 
Ex. 1. 

86. When we have given the two implicit functions, 
u - F{x, ^, 2 . . .) = 0, V -f{x, y, 2 . . .) = 0, 



IMPLICIT FUNCTIONS OF VARIABLES. 129 

of the same variables, we should have at the same time 
du = 0, c?v = 0, from which can be determined the differentials 
of any two of the variables considered as implicit functions 
of all the others ; and, in general, if the relation between 
the n variables, ic, ?/, 2 . . . , is expressed by the m equations, 
2^ = 0, 17 = 0, t6;=:0..., we should have at the same time the 
m differential equations, 

du =i: 0, c?i; =: 0, (iw; := . . ., 
and, by means of these, could determine the differentials of m 
variables regarded as functions of all the others. 

If the number of variables exceeds only by 1 the number 
of equations expressing the relations between them, one of 
the variables alone can be independent ; and we may find the 
differential co-efficients of all the others regarded as functions 
of this single variable. 

Let us have n equations, 

u^ = E,{x,y,z...t) — 0, 



^n = K {x,y,z ...t) — 0, 
between the n -\-\ variables x, y^ z . . .t. 

Differentiating all of these equations with respect to x 
taken as the independent variable, we have 

duy dui dy du^ dz dui dt 

dx dy dx dz dx dt dx ' 

du,^ du^ dy du., dz du., dt 

~dJ "^ 'dii dx + ~dz dx^' ' "^~drd:vr^^' 



dic„ du,, du du,, dz du„ dt 

7^7 "*" Hy dx ^~drdx'^'''^~dt 7/7 ~ ^'• 

Tlioro are n of those dilfcrcntial oipiations involving the ?t 
17 



130 DIFFERENTIAL CALCULUS. 

required quantities, ~^ ;j~ * • • 'j~j wt^ich may therefore be 
determined. 

87, When the variables enter the function in certain 
combinations, the results of differentiation take special forms, 
and peculiar relations exist between the partial differential 
co-efficients, depending on the manner in which the variables 
are combined. We shall first consider the case of homogene- 
ous functions. A function is said to be homogeneous when 
all the terms entering under the functional symbol are of the 
same degree with reference to the variables. Thus 

F {x, y, z) = ax^- -\- hy'^ -\- cz"- -\- 2eyz 
is a homogeneous function of 2 dimensions, and 

F{x-, y)=.^ 

is a homogeneous function of dimensions. A property of 
such function is, that, if all the variables are multiplied by 
the same quantity, we obtain for the result the original 
function multiplied by this quantity raised to a power whose 
exponent is the number denoting the dimensions of the 
function. Therefore, if F{x^ y, s . . .) is a homogeneous func- 
tion of a dimensions, and t denotes a new and independent vari- 
able, we have 

F{tx,ty,tz. . .) = t''F{x,y,z . . .). 

Put tx^u^tyz^v^tz^^w. . . , then 

F{u,v,io. . .) = t''F{x,y,z , . .) ; 

and differentiate both members of this equation with respect 
to t : the result is, 

dF du , dF dv , dF dw .„ a -n^ n 

~7 Ti-\- -1 t:-\--i iT ' ' ' — (^t Fix, y.z . . .). 

du dt ^ dv dt^ dw dt ^ '^' ^ 



IMPLICIT FUNCTIONS OF VARIABLES. 



131 



But 

therefore 



du 
di 



dv 
di 



dw 
~di 






dF ^ dF ^ dF ,a 1 XT, X 

X ^r~-\-y -, — Us---... = a^«-i Fix, y,z . . .). 
die "^ dv ^ dw ^ '^' ^ 

Now, since t is entirely arbitrary, make t^zl] then 

, dF dF dF dF 

uz=x, V :=^ y, to z=z z . . ., and -^- = -y-, -^ :=z — - . 
' ^' ' du dx dv dy 



whence we have 



dF ^ dF , dF 



dx 



dz 



z= aF{x, y, 



The first member of this equation is the sum of the products 
obtained by multiplying the partial differential co-efiicients of 
the function, each by the variable to which it relates ; and the 
second member is the primitive function multiplied by the 
number denoting the degree of the function. 

If the function is of degree, 

dF ^ dF ^ dF ^ 

Ex. 1. F{x, y, z) = ax'" + hy' -}- cz' + 2eyz + 2/zx + 'Igxy, 
^^- - 2ax 4- 2fz + 2gy, ^^^ = 2hy + 2cz + 2gx, 



dF_ 
dz 



= 2cz + 2ey + 2fx, 



and a — 2 : therefore 

(2ax + 2/2 + 2(/-7)x 
+ (2% + 2^'. + 2../.r)// 
+ {2cz^-2cy^2fx)z 

an identical equation. 






132 DIFFERENTIAL CALCULUS, 

Ex.2. F(x,y)—-, -,- = -7 -7— =—-2? 
^ '^^ ij' dx y d.y 2/ 

a = : therefore 

dF , c?i^ X xv x X ^ 
dx ' "^ dy y y y y 

SS, Let us next take the case of the function of the alge- 
braic sum of several variables, x, y, z . . . If the function be 
u := F{x =ti ^ rfc 2 =ti . • '), and we put icdc?/=bs± . • - z=tj 
it becomes u ^= F{t). 

Now, if the original function be differentiated with respect 
to x,y^z . . . separately, we shall have, by reason of the equa- 
tion u — F{t), 

dF _dF dt dF _dF dt dF __dF dt 
dx dt 5:c ' dy dt dy ' dz dt dz 

But the equation x^y ^z^ - - • z=zt gives 

dt _ ^ _ .dt ^ ,dt 

dx dy dz 



therefore 



dF_^dF_^dF_^ 
dx~ dy dz 



that is, the partial differential co-efficients of the function are 
numerically equal. 

Ex. 1. « = (^ + y)», ^ = | = n(a. + 2/)«-'. 

Ex. 2. u = {x- yy, J = - ^ = n(.r - yf-K 

. du dvb 1 

Ex. 3. u— l\/x-\-y 



Ex. 4. u — l\/ 



dx dy 2[x -\- y) 
du du 1 



X 



dx dy 2{x — y) 



SECTION IX. 

SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE IN- 
DEPENDENT VARIABLES, AND OF IMPLICIT FUNCTIONS. 

89, In Sect. IV., rules were investigated for the succes- 
sive difFerentiation of explicit functions of a single variable. 
We now pass to the successive differentiation of functions of 
many variables, all of which, at first, will be supposed inde- 
pendent of each other. 

By Art. 81, the total differential of a function of several 
variables is the algebraic sum of its partial differentials ; and 
it is evident that each partial differential co-efficient is, in 
general, a function of these variables, which may be again 
differentiated with respect to a part of the variables, or with 
respect to the whole of them. These operations give rise to 
what are called partial and total differentials, and differential 
co-efficients of the different orders. 

00, If ?i = F (x, ?/, 2 . . .) be a function of the independont 
variables x, y, z . . . , then du, d'hc, d'hi . . . d" u . . . , standing 
by themselves, will denote the first, second, tliird . . , ;/*'* total 

differentials of the function. is the first partial dillorontial 

ax 

co-efficiont of a taken with respect to .r ; anil dx, or (/..//. 

dx 

IS the corresixuuiinir iKirtial ilillerontial. _:= , is 

dx\dx/ i/x- 

the second partial dilVerential co-eniciont Avith respect to x: 

loo 



134 DIFFERENTIAL CALCULUS. 

and the differential corresponding to it is — — dx^^ or d\u. 
— ( — jrzi- — — ^^ ^^ second partial differential co-efficient, 



dy \dxj dydx 

taken, first with respect to x, and then with respect to y ; 

d'*"U 
and the differential answering to it is dydx^ or dyd^u. 

CLIJCLX 

— ( — ^ ) zzz is the third partial differential co-efficient 

dy \dx^ J dydx'^ 

of the function obtained by differentiating twice with respect 

to x, and then once with respect to y ; and to this we have 

d^ u 
the corresponding differential ^r— ,— 2 ^2/^^S ^^ <^\<^yU) which 

may also be denoted by d^^yU. In like manner, the notations 

d^u d^u 

d^^dz' SF^di '^''^y'^'' '^'''^^<^'^' ^''^^«' ^'°"'d indicate 

four differentiations : one with respect to z, one with respect to 
y, and two with respect to x. From what precedes, the significa- 

tion of the notations , , , ^- — , ■ , ^ 7 „ 7 „ — dx^^dy^'dzP. 

dx'"- dy"" dzP . . J dx"" dy"" dz^ . . . ^ ^ 

d^d^d^^ .. . u, d'l^r^ yn J '['u, will be readily understood. 

The remarks in Art. 81, in reference to partial differential 

co-efficients of the first order, are equally applicable to those 

of the higher orders. Keeping in view the principles there 

laid down, there will be no risk of confounding any order of 

partial differential of the function with the total differential 

of the same order. Thus, in , , , the d'^u is always associ- 
ated with dxdy written below it; and in this way the con- 
struction of the expression indicates both the character of 
the differential co-efficient, and the variables with reference 
to which it is taken. 

It is often convenient to attach to the symbol of the func- 



FUNCTIONS OF TWO OR MORE VARIABLES. 135 

tion the characters by which are denoted the order of differ- 
entiation, and the variables involved in the operation. Thus, 
K (^; 2/; 2 • • • ); K {x, y, z . . , ), F'^', ^ {x,y,z... ), have re- 

. , ,, • -n du d}u d'^u 

spectively the same signmcance as — , T~~T' "> ii.i i ^^ 

cLx diXiCLy ctx (Xy 

above explained. 

91. Before proceeding farther, we must prove, that, in 
whatever order in respect to the variables the differentiation 
of a function of many independent variables is effected, the 
result is always the same : that is, if i^ = i^(x, ?/, 2 . . .) is to 
be differentiated m times with respect to ic, and n times with 
respect to ?/, the result is the same, whether we perform the 
m ic-differentiations, and then the n ^-differentiations, or re- 
verse the order of differentiation in respect to x and y ; or 
perform first a part of the m ic-differentiations, then a part 
of the n ^/-differentiations ; and so on until the whole of the 
m and n differentiations are effected. 

This principle may be demonstrated as follows: Take the 
function u ^= F (x, y, z . . .) of the independent variables 
x,y,z . . . Suppose, in the first instance, x to be variable, and 
all the other variables constant, give x the increment h^ and 
develop by the formula of Art. 61 ; then, in the result, suppose 
y to be variable, and all the other variables, including .r. to bo 
constant, give y the increment A*, and develop the terms by 
the same formula. The final result will be the same as that 
we sliould have reached by giving the incM-emonts to .r and // 
simultaneously. 

Changing x into x -\- 7i, then, Art. CA, 

F{xJrJh !/, '':...) = ^(•^•. //. .-:...)-[- h F; ^x, //, z . . .) 

+ ^'-F;:^x^f),h,y,z...).., ^\); 



136 



DIFFERENTIAL CALCULUS. 



in which F'J, (x-{- 6^h,y, z . . .) is a function x,hyy,z . . .^ which 
remains finite when h = 0. 

If in (1) we change y into y -\-h, the first member becomes 

F{x + 7i,y + k,z. ,..)', 
and the terms in the second member become respectively 
F{x, y + k,z...) = F{x, y,z.,.)-{- hF'y {x, y, z . . ,) 

hFl {x, y +h, z. . .)= liF'^ (x, ?/, s . . .) + likF'^yix, y,z...) 
+ ^F:;.ix,y + e,k,z.,.) 

^^F'\A^ + e,h,y + h,z,,.) = ''^F';A^^O,]i,y,z...) 

+ ^'^F';:^{x^d,li,y^e,h,z,.,), 

Making these substitutions in Eq. 1, we find 

F{x, y, z...)+hF',{x,y,z..,) 
+ hF;(x,y,z...) 
+ hkF^'^{x,y,z...) 



F{x + h,y^k,z...)=z 



+ ^F'j2 ixi-d,h,y,z...) 

+ ^-^FZMy^o,k,z...) 

j^^^F';:^{x^d,h,y-^rd,k,z 



If we begin by giving y its increment, we shall have the 
equation 

F{x, y~\-k,z...):^ F{x, y,z...)+ kF^ {x, y,z...) 

and in this, giving to x its increment li, and developing the 
terms as was done above, we have 



FUNCTIONS OF TWO OR MORE VARIABLES. 13' 



F{x^h,y-^rh,z.,.)=z 



-^-hF'yix.y, z..,) 
hhF';^{x,y,z...) 

^^F:.M-\-d,h,y,z...) 
+ ^^F';.{x,y + d,h,z...) 

+ ^^F-.{x^O'h,yr^...) 
^-^^F';A^-^d'ni,y-\-0,k,z.,.). 

It is to be observed, that, in the several preceding equa- 
tions, the factors F!^^ {x -f- d^li, y, z . . .), F'^.'y^ {x, y -\- O.J^, z . . .), 
&c., of terms in the second members, remain finite when 7i and 
k, separately or together, become zero. 

Equating these two values of F {x -{- h, y -\-k, z . . .), sup- 
pressing terms common to the two members of the resulting 
equation, and dividing through by 7ik, we have 

k 
Fj; (x, y,z...) -{-^F^:^, {x,y-^ d,k, z...) 

+ lF:::,^{xJr^Al/ + 0,k,z) 

This equation must be true, whatever the vahies oi' h and A*. 
Make Ji — 0, k ~0; then 

F:'y{x,y,z,..)z=zF';^.{x,y,z) {'2). 
The first member of this equation is the second jnirtial dif- 
ferential co-efTicient of the function obtained by dilVoronti- 
ating, first with respect to .r, and tlien with respect to // ; the 
second member is the second partial dilVerontial co-ctlicicnt 

IS 



138 DIFFERENTIAL CALCULUS. 

which comes from differentiating^ first with respect to y, and 
then with respect to x. It is therefore immaterial in what 
order the differentiations are performed. 

This theorem being demonstrated for derivatives and dif- 
ferentials of the second order, it can easily be extended to 

dfj 

those of any order. Suppose we start with ■ — . Whether 

dz 

this be differentiated, first with respect to x, and then with 
respect to y, or we invert the order of differentiation, the re- 
sult is the same by what has been proved. So that 

d^'u _ d^u 
dxdijdz dydxdz 

But the order of differentiation with respect to 2, and either 
x or y, may also be inverted ; and therefore 

d^u d^ic d^u 

dxdydz dydxdz dxdzdy 

and generally, for the function u = F{x,y, 2 . . .); 



dx^'dy'^dz^ dy^'dx'^dz^ dx'^'dz^dy'' 
Ex. 1. uz=z 



x' — y'- 



x^^if 
du 4:xy'^ du 4ic^y 



dx-{x'^yy' dy {x' + y^f 

d^u Sxy{x'^ — y''-) d-^u Sxy{x'^ — y-) 

dydx {x^ -\- y'^Y dxdy {x'^ -\-y'^f 

X 

Ex. 2. i^ = tan.-^-; 

y 

du y du X 

dx x'^ -\-y'^^ dy~ x^ -\- y'^ 

d'^u _ x'^ — y'^ d'^-u _ x'^ — y^ 
dydx ~ {x' -\- y^Y^ dxdy ~ {^x^ + y'^Y 



FUNCTIONS OF TWO OR MORE VARIABLES. 139 

92, The differential of an independent variable is a con- 
stant, and therefore the differential of the differential of such 
a variable is zero. Now, since the differential co-efficient of a 
function of the independent variables, x, ?/, 2, with respect to 
one of these variables, is a new function of ic, ?/, 2, and the corre- 
sponding differential is obtained by multiplying the differential 
co-efficient by the differential of the variable to which it re- 
lates, it follows that, in subjecting such differential of the 
function to further differentiation, we may set aside the differ- 
ential of the variable as a constant factor, and operate on the 
differential co-efficient alone ; restoring in our final result the 
constant factors set aside: thus, if ^^ = i^(x, y, 2), in which 
X, y, and z are independent, then 

d^u^=: jpj {x, ?/, z)dx ^ ^— dx^ 
dyd^u = dxdyF'^{x,y, z) = dxFy^{x, y, z)dy 
=Fy^ {x, y, z)dxdy ~ ^^-^- dxdy, 
d.dyd^u = dxdyd,Fy^(x, y, z) — dxdyFl'y,.{x,y, z)dz 
— F^y^{x, y, z)dxdydz — -^ rr^ dxdydz^ 
and, generally, 

d^dyd^uzzi 1< p n m (-»^, y, z)dx"'dy"dz^' 

= -, ^--.^dx"'dy"dzP. 

dx"'dy"dz^ '^ 

03. By Art. 81, the first total different ial oi' tlio function 

u =: F{x, y, z), of the variables x,y, z, is 

, dio J die J . du J . ^ 

ax dij (/;.' 

Taking tlie total diiroi-ontial o[' c\\c\\ o{^ the partial din'oreutial 

rt, . , du, du du , 

co-eilicionts ,',',, wc have 
dx dy dz 



140 DIFFERENTIAL CALCULUS. 

-. fdu\ dP-u , , d?u , d'^u , 

n (du\ __ d^u , d^-u , c?% 

, fdviX d'^-u , (i-z^ , (i% 

\(i2/ d%dx ^zdy dz^ ' 

and therefore 

'^''-5^'^'' ^"^^^ ^d^ ^ d^ ^ 

Proceeding with (2) in the same manner that Ave did with 
(1), we should get the third total differential of the function; 
and so on. 

For the function u^^F{x^y) of the two independent vari- 
ables X and y^ the successive total differentials will be 
7 dii ^ , du , 

dij d'^ii dii 

d'^u = -^—rdx'^ -\- 2 -—— dxdy + -^. c?2^ 
dx^ dxdy dz^ 

d-u = ^ dx'^ + n -JL::^dx-'dy 
dx"" ~ dx^'-^dy ^ 

. n(n — 1) d^'u , „ 9/70, 
^ 1.2... («-l) dxdy"--^"''"^y 

+ dr'^y ' 



FUNCTIONS OF FUNCTIONS. 141 

the law of the co-efficients being the same as that in the de- 
velopment of (1 + xy. 
Ex. 1. u^=. xyz^ 

du =: yzdx -f- xzdy -\- xydz, 
d'^^u^^l {xdydz -\- ydxdz -\- zdxdy)^ 
d^it = Qdxdydz. 
Ex.2. u = {x'' + y^Y, 

du X d'^u __ y'^ d^u Sicy^ 

dx (a;2 + 2/')' dx"- {x''-{-y''f dx^ (^2_^^2)l 

du y d^u x' d'^u 3x-y 

dy {x' + y^'f dy' {x'^-y'f dy' {x'^y'f 

d^u xy d'^u _ 2/(2aj^ — ?/') 

dxdy (^2_|_^2^l ^-^l^y (^2_|_^2)l 

dxdij'^ (x'^A-y-^f 

d'u = (- ^x2fdx^-\- 2>y {2x'' — y"-) dx'-dy 

+ 2>x (2^/2 _ x2) dxdy"- — Syx'^dy 
Ex.3. u=:e"''+^''', 

du ^ , , d'U 

dy dy- 

d'u , _,, 
dxdy 

d'u — {a'dx' -f 2(ibdxdy -f- b'-dy-)e"-'^-^^y 
= (adx^l>dy)\"''' +''!'. 
04. If, in tlio function s = I\>(, i\ ir), u, r. ami ;r are 
functions of the indc})on(lont variabK^s x, y, and ,:. wo liavo a 



{x' + f)i 



142 DIFFERENTIAL CALCULUS. 

case of a function of functions of independent variables ; and 
the first total differential of s is 

ds ds ds . 

But since u, v, and w are all functions of x, y, and z, tlie par- 
tial differential of s, regarded as a function of x, is (Art. 82), 

ds , ds du ds dv _ ds dw ^ 

—-dxzz^- — — dx -\- - — —dx-{- — dx : 

dx du dx dv dx dw dx 

ds ds dio ds dv ds div 

so also -r^y — :ri~^y~^i~~r^y~^zr':r^y' 

dy du dy dv dy dw dy 

ds ds du, ^ ds dv ds dw 

and —- dz :=— — -— dz -\- - — -- dz -\- 1- dz. 

dz du dz dv dz dw dz 

The total differential of u is 

^ du ^ du ^ du ^ 
du :=. —- dx -\- —- dy -{- -z-dz ; 
dx ^ dy ^ ^ dz ' 

and, for the total differentials of v and w, we have like expres- 

sions: therefore, by substituting these values of -—(ix, — dy, 

(XX cty 

ds 

~dz in (1), and uniting terms, we have 
dz 

ds ^ ds ^ ds ^ 
ds=z—-du-\-~~dv-{--z— die. 
du dv dw 

A second differentiation would give 

79 ^^^ 7 O <^^<^ T O <^^S 7 O r. (^^S 7 7 

a-s = -7—^ du^ 4- -—- dv^ 4- -7— ^ dw^ 4- 2 -— — - dudv 
du" av" dw^ dudv 

^ d^s 77 ^ ^^5 7 7 ^-5 70 

+ 2 - — —- dudw -}- 2 dvdw -\- ^j-d^u 

dudw dvdw du 

ds ,„ ds _^ 

4- -— d- V -\- --- d-w : 
dv dw 

and from this we pass to d'^s^ and so on. 

The general rule is, then, to differentiate as if u^ v, w, were 
independent variables, and substitute in the results the values 



HOMOGENEOUS FUNCTIONS. 143 

of du, dv, dw ; d'-u^ d'^v^ d'^w . . . , derived from the equations 
giving u, V, IV, in terms of the independent variables x, y, z. 
If 10 =i ax -\- by -\- cz -\- d, V ^= a'x -|- h'y -j- c'z -\- d', 
lu = a"x -\- h"y + c"z + d" , 
are the expressions for u, v, lu, in terms of x, y, z, these func- 
tions of the first degree, with respect to the independent vari- 
ables, are said to be linear. In this case, we should have 

dhc z= 0, d'v = 0, dhv — 0, d^u = • • • ; 
and the successive differentials of s = F^^u^VjW) would then 
have the form of the successive differentials of a function of 
three independent variables : thus 

d'^s c?^9 d'^s d'^9 

d'^s — -^-r, du^ 4- -—■ dv'^ -{- -y-T, dio'^ 4- 2 - — — dicdv 

dio^ dv^ dw^ diidv 

+ 2 -- — — dudw -(- 2 -^— ,— dvdw. 
dudw dvdw 

Ex. 1. s ^= F{u,v), u = ax -\-hy -\- c, v = a^x-{-b^y -{- c^, 

du" du"~hlv dv" 

Ex. 2. s=zF{ u) F{v), u = ax + hy -{-c,v=: a'x -}-I/y-{-c% 
ds = F' (ic) F{v) du + F{u) F' {v) dv, 
dh = F''{u) F(v) du' + 2F{u) F' (v) dudv + F{u)F'[r)dc- 

d"s = F^"^ (u) F(v) du" + nF^"-''{u) F'[v) du"-' dr + • • • 

+ nF\ic) F^"-'' (v) dudr"-' + F[u) F'\r) dr" • • • 
0^, If the function 5 = F{x, y, z) of the indopondont varia- 
bles, x, y, z, is homogeneous, and oi' a diuionsions, thou, bv Art. 
87, 

aF{x,y,z)=x- -\-y + .: , . 
dx ay dz 

It may bo shown that similar rc^lal inns oxist botwoon ilio func- 
tion and its dillcrontial co-onicicnts lU' the hiuhor ordors. 



144 



DIFFERENTIAL CALCULUS. 



Since the function is homogeneous, if we change x, y^ and z 
into tx, ty, tz, and make ic ^tx, v ^=^ ty, lu =: tz, we have 

F{u,v,w) = t''F{x,y,z). 



Differentiating this twice with respect to t, observing in the 
second differentiation that | (J), ^(J), |(JJ), are each 
zero, we find, 



dF du , dF dv , dF diu , , -rr , 

-7 7r + ^--7.-f-7 TT =at''-^F(x,y,z 

dii dt ' dv dt ^ dio dt v ?^7 > 



d^dv^ d'^F dv^ d'-F dw'- 

d'F du dv d-F du div 

"* dudv dt dt ~^ "^ dudw dt dt 
d-F dv dio 
^ dvdio dt dt . 



^a{a-l)t"-^F(x,y,z). 



But -^ = cc, — =r ?/, — ~ = 2/ and, if ^ — 1, the second partial 

differential co-efficients of the function with respect to u, v, w, 
become the second partial differential co-efficients with respect 
to X, y, z, respectively ; and hence the last of the above equa- 
tions becomes 



d'-F 



dx 



,+y' 



d'F 



d'F 



dy'- dz^ 

, _ d'^F ^ ^ d'F ^ _ d'F 
4- 2xy -^r-T- + 2x2 -^ — — -\- 2yz ^ ^ 
^ ^ dxdy dxdz ^ dydz} 



= a{a~l)t^-'F{x,y,z). 



By a third differentiation, we should get 



d'F 



d'F 






d^ 
dz' 
d'F 
dxdy^ 



...i 



— a(a-V)(a-'r)t^-'F{x,y,z), 



DIFFERENTIATION OF IMPLICIT FUNCTIONS. 145 

Example. F{x, y, z) .— ax"^ -{-hy'^ -}-cz^-\- "Iq^xij + Ifxz 4- Igyz^ 
d'^F ^ d"^F ^^ d^'F ^ 

(i^rc??/ ' dxdz *^' o??/o?2 *^ 

+ 2:^2-"^- +9,,i^l^ ^ 
(i^cZs "'""'^ (iz/c/z ~ 

2 (a^2 ^ ^^2 ^ c^2 _^ 2eicy + 2fxz + 2^2/2) = 2i^(:z;,?/, z). 
96. To express the successive differential co-efficients of 
implicit functions, take the function u =:F{x,i/) = 0, in which 
y is implicitly a function of x; then, by Art. 84, 
du da dy _ 
dx dy dx~ 
The first member of (1) is another function of x and y, which 
denote by v ; whence i' = 0. Differentiating v = as we did 
^^ z= 0, we have 

dv dvdi__ 

~r ^.. ^v. — ^ v-^.M 



dx dy dx 

dv d^u d'hc dy du d-y 

dx ^j?'^ dxdy dx dy dx- 



1 , uu ct M/ ii CO ay 



, c/v _ dric d^ u dy 

dy 7irdy dy' dx 

dv dv 
These values of -' -, , , substituted in (2), o-ivo 
dx dy \ '■ ^ 

d'u ^ d'^io dy d'U /dyV du d'-y _ .. 
dx'^ '^ dxdy dx dy'^ \dx/ dy dx- ~ '^ ' 

From (1) and {?.), wo find llio valuos oi' ';'^ '^'-i. Ens. 1 and 

dx i/x' 

3 are called dinbriMitial or derived (M]ua(ions o( the first and 
second orders respeclivoly ; anJ, with relorenoo to thorn, 
io=F{x,y) — is the primitive equation. 

19 



146 DIFFERENTIAL CALCULUS. 



The above process is somewhat simphfied by putting -~:=zp • 

doc 



then 



du , du ^ ,^.. 



^ dv d'^^u d'^u du/dp 

dx dx'^ dxdy dy\dx 

dv d'^u d'^'u J. du/dp 

dy dxdy dy^ dy\dy 

These values in (2) give 

{d'^u du /dp\ d^-u ) c?% du/dp\ d''-u^ 

W^^\d^y) dxd^\^'^d^ ' 

" S+^S,^+S''-+||(|)-+(l)}=«- 
""* (|)^+(l)=S=2<-'*-«^>-"""" 

or ^^^ , ^ ^% dy d'^u /dy-^ du d^y _ ^ 

dx'^ '^ dxdy dx dy'^ \dx) dy dx^ 

We call attention to the notations f — ], ( — i, -^, by re- 

\dxl \dy) dx' ^ 

marking that p is generally a function of x and y ; and that 

d'p\ (dp" 



^, ^ _ ,, are the partial differential co-efficients of this func- 
dx) \dyl 

tion, the first with respect to x^ and the second with respect 

to V : whereas -;^ is the differential co-efficient of ip with re- 
^ dx ^ 

spect to x^ p being considered^ as it is, a function both of x 
and of a function of a function of x. Thus suppose, that, by 
solving the primitive equation F{x., y) = 0, we find y =/(x)y 
then 



DIFFEUENTIATION OF IMPLICIT FUNCTIONS. 147 

Suppose also^ that, without solving the primitive equation, we 
find p = cp {x, y) =q^ fx,f(x)j ; then 



|) = .r;(-,y), 



[^) = <r'A-,v)-- («)• 



But, by Art. 82, 



This points out the necessity of distinguishing, in certain 
cases, partial differential co-efficients, such as those of p in 
Eq. a, by the parenthesis, or some other mark, that, in 
the course of an investigation, they may not be mistaken for 

others, as -^ in Eq. b, of the same form, but having a differ- 
dx 

ent significance. 

dx'^ 



d "9/ 
The value of -j~ deduced from Eqs. 1 and 3, or from V 



and 3^, is 

d'^^u /du\} d'^u dvb dii dhv /duV 

dhj dx'"\d(i) dxdy dx dij di/''\dx/ 

The expressions for tlio higher orders oC dllVerontial co-oi1i- 
cients of implicit functions are so conii)licated, and so littlo 
used, that it is unnecessary to ]H-()coed farther \\\i\\ this divi- 
sion of the subject; but we will conchido it by giving t ho 
differential equation of the third ovdcv oi' iho implicit func- 
tion y of the variable x, given by the equation u := F{x,y) := 0. 
This dillerential ocpiation is 



148 DIFFERENTIAL CALCULUS. 

d^u ^ d^u dy d^u /dy\^ d^u fdy\^ 

dx^ dxHy dx dxdy'^ \dx) dy^ \dx) 

^/ d'^u d'^u dy\ d'y du d'y _ ^ 
\dxdy "^ dy'' dx) dx'' ^ dy dx^ ■ " ' ^ ^' 

97, Suppose we Lave given the two simultaneous equa- 
tions 

u=F{x,tj,z) = (1). 

v=/(x,if,z)=0 (2). 
It is theoretically possible, by combining these equations, to 
eliminate either variable, and get an equation expressing the 
relation between the other two from which the successive dif- 
ferential co-efficients of one of these regarded as a function 
of the other might be obtained. But without effecting this 
elimination, not always practicable, we may proceed as fol- 
lows : — 

Suppose x to be the independent variable, and differentiate 
(1) with respect to x; then (Art. 85) 

du du dy du dz __ ^ ,^. 
dx dy dx dz dx 

In like manner, from (2), 

dv dv dy dv dz _ „ ., 
dx dy dx dz dx 



(5), 



From (3) and (4), we find 




du dv 


dv du 


dy dx dz 


dx dz 


dx ~ du dv 


dv du 


dy dz 


dy dz 


dv du 


du dv 


dz dx dy 


dx dy 


dx" dv du 


du dv 


dz dy 


dz dy 



(6). 



DIFFERENTIATION OF IMPLICIT FUNCTIONS, 149 

The first members of (3) and (4) are functions of x, y, z; and, 
by differentiating them with respect to x, we have 

d'^u ^ d'^u dy d"U dz d'^u /dy\2 d'^u dy dz 

dx^ dxdy dx dxdz dx dy'- \dx) dydz dx dx 

d'-ii fd'^y, du d-y diod'^z __ 
'^ d^\dx) "^ dy dx"' '^dzd^'~ ^" ^^ ^' 
and 

dx'^ dxdy dx dxdz dx dy'^\dxj dydz dx dx 

d'^v /dz \2 dv d'^y ^ dv d'^ _ ^ ^. 

'^dz'\dxj~^d^'dx'^dzd^'~ '" ^ ^' 

From (7) and (8), by substituting in them the values of 
J^ , J- , in (5) and (6), we may deduce the values of --,^ 

d'^z 
and ^— ,. They may also be found directly by differentiating 

(5) and (6). 

08, For an application of the methods of successive differen- 
tiation, suppose we have the single relation u ^=zF{x,y, z) = 
between the three variables x, y, z; then z may be consid- 
ered as an implicit function of the two independent variables 
x,y. 

It is reqiured to find the first and second orders of tlio par- 
tial differential co-efficients of z with respect to x ixud y witliout 
solving the equation u = F{Xfy, z) — 0. 

The first partial derived equation with respect to x i? 

(Art. 85), 

dfo , di{, dz ^ , . 
dx dz dx 

and tliat with respect to ?/ is 

(/// dz ay ^ 



150 DIFFERENTIAL CALCULUS. 

in which -^ . -j- , ^-. are the partial differential co-efficients 
ax ay az 

of u, taken on the supposition that the variable x, y^ or 2, to 

Avhich they separately relate, alone varies. 

Eqs. 1 and 2 will give — , -=~. Differentiating (1) with re- 
ax ay 

'« 
spect to X, and (2) with respect to y, and also either (1) with 

respect to y^ or (2) with respect to x, we get 

dx"'^ dxdz dx^dz^\dx)~^dz dF'~ *^^^' 

dhi ^ d}u dz d'^-u ldz\^ dvb d'^'z __ ,, 

^"^ d^zd^^d¥\^)~^dzdY^~~ ^^' 

d'^u . d'^u dz d'^u dz _,d^^u dz dz du d'^^z _ ,^, 



dxdy dzdx dy dzdy dx dz^ dy dx dz dxdy 

and from the five Eqs. 1, 2, 3, 4, and 5, we can deduce 

dz dz d^-z d'^z d'^z 



dx dy dx'^ dy'^ dxdy 

Ex. 1. Given y^ -\- x^ — ?>axy = 0, to find the value of -, ^.,. 
The first differential equation is 

and the second, 

Substituting in (2) the value of ~- taken from (1), we find, 

ax 

after a little reduction, 

y—x''){y'^'-ax)- 

-^2x{y^ — axf = ()'. 



d'^-y 

(^2 _ ^^)3 _| _ 2a (a?/ - a;2) (y 2 _ ax) + ^ay - x') 



whence 

d^y _2a{ay — x'-) (y'' — ax) — 2y{ay — x'^f — 2x{y^ — ax)\ 

dx^ ~~ ~~ (?/2 — axf ' 



DIFFERENTIATION OF IMPLICIT FUNCTIONS. 151 

and this^ after performing the operations indicated in the nu- 
merator of the second member, and reducing by the given 
equation, becomes 

dx^ {y^ — Gbxf' 

Ex. 2. Given I'^cP'x' ^ a'c^y'- ^ a}l''z'' — a'h''c' — ^,\o^\i^ 
d^z d^z d'^z 
dx'^^ dy'^^ dxdy^ 

dH __ c\aH'' 4-c^x^) dH _ _c\hH'--\-c''y'-) 
dx"" ^'¥~ 'df'~~ "&V ' 

d'^z c^xy 

dxdy a^^h^z'^ 



SECTION X. 

INVESTIGATION OF THE TRUE VALUE OF EXPRESSIONS WHICH PRE- 
SENT THEMSELVES UNDER FORMS OF INDETERMINATION. 

99, It sometimes happens that the expressions nnder con- 
sideration assume, for particular values of the variable or 

variables involved, some one of the forms ^, i— ? X oo, 

O*', Qo^, it 1"°; 00 — oo, called forms of indetermination, though 
the value of the expressions may be determinate. Our object 
now is to establish the rules by which may be found the true 
value of an expression which reduces to any one of these forms. 

100, Of the Form -^- This form can only result from a 

fraction in the numerator and denominator of which there is 
a common factor, w^hich factor becomes zero for the particular 
values of the variable or variables which reduce the expression 

to - . Thus take the fraction — ^^ v- , in which F and 

Q {x-af ^ 

may or may not be functions of x ; but, if they are, they do 
not contain the factor x — a, and therefore do not become zero 
when x = a. If in this fraction, as it stands, we make x =: a, 

it takes the form - ; but if, before giving x this value, the 

. P 

fraction be written -^ (^ — «)"' "? it is seen that the true value 

P 

of the fraction for cc = a is if m yn,cc if m < n, and -p. 

if m = n. This suggests the following rule for the evaluation 

152 



INDETERMINATE FORMS. 153 

of expressions whicli take this form ; viz., discover, if possible, 
the factors common to the numerator and denominator of the 
fraction, and divide them out. What the result reduces to by 
giving the variables their assigned values is the true value 
of the expression. ' 

Example, ^-^^^f^;^^ = J ''^'''' " = "^^ ' 

^ x^—2.x'' — ^x-\-Q>~ (0^2 — Z)\x—2)~x-2 

1 + ^^ ^ /o 

— — — 7- for x = \i. 

Many cases of the form - may be treated as follows : — 

Take the fraction -^ ~ — , which becomes - when a*= a. 

Make x := a -\- h; then 

(a 4-h ~ a)^ id' A« ^ , 
^ ^- 1 — - — — when 

(a2 + 2a7i + h'' - a'')i h^ (2a + h)^ (2a + 7i)^ 
/i = 0, which corresponds to x = a. 

. 1 ,T r , • \^x — Va 4- Vx — a _ 

Also the traction — ^^^~ — - for x — a; mak- 

Vx'-a' 

ing X =za -\- h, 

Va + A — Va -\- Vh _ Va -f h — [Vci- — >/h) 

V^aJr+'Ji/^ ~ \/2a/i + 7i^ 

Multiplying numerator and denominator oi^ this by \^a-\- It 

-f (Va — Vh), wo find 

2 Vah 1 

(2a/^ + Jr)^ ({a + h)^ + (\^a - \0i)\ ~ V2a 

for A = 0, after diviiling out ihc coininou factor Ir- . 

The examples already givcMi have boon sohed by common 
algebraic (ranslbrmalions; but. most of the cases Avhicli present 



154 DIFFERENTIAL CALCULUS. 

themselves can be more easily solved by means of the differ- 
ential calculus. 

Fix) 
101. Suppose the fraction to be ^ ' and that both F(^x) 

and f{x)j as also their successive differential co-efficients up 
to the (n — 1)*'^ order inclusively, vanish for x:=za; then it 
has been proved (Art. 56) that 

F{a-i -h) __ F^-\a-^dJi) 
/{aJrh) -f"\a + 0h)^ 
and consequently, by making A = 0, we have 

F{a)_F^''\a) 
f{a)- r\ay 

Hence, to obtain the true value of the vanishing fraction 

Fix) 

-—— when x-^a, form the successive differential co-efficients 

of both terms of the given fraction until one is found, whether 
of numerator or denominator, that does not vanish for x = a; 
and take the value, when x=^ a^ of the fraction whose terms 
are respectively the differential co-efficients, of the order of 
that thus found, of the corresponding terms of the given 
fraction. 

If one of these differential co-efficients vanishes, the value 
of the fraction will be or oo , according as it is that of the 
numerator or of the denominator ] and it will be finite if the 
first of the differential co-efficients that do not vanish is of 
the same order in the two terms of the fraction. 

Ex.1. ^ . ^ - = liox x = ^. 

sm. X 

F{x) =ze^ — e-'% F\x) — e^ + e"^, f{x) = sin. ic, 

f\x) = COS. X, 

'F_ix)\ _/F'{x)\ ^/ e- + e-' \^ ^^_ 
'{^)A = o \/'(a:)A = o \ COS. a; 



Ex.2. 



INDETERMINATE FORMS. 155 

'x — sin. x\ /I — COS. x\ /^m.x> 



COS. X\ 1 



6 Ja: = i) 6 



Ex. 3. . ^- \ - n\ 



102, Form ^- If the two functions F{x), f{x), become 



infinite for x =. a, the fraction -^^-4 reduces to — • But in 

/{X) OO 



this case the fraction may be put under the form jZlk^, which, 

for X z:^ a, becomes -, and may therefore be treated by the 
preceding rule. Thus 

1 -■^-^^^-, 

whence -,./ , = ,;, / 4': ^^d the true value of the ratio -, '-- =:: ^ 
/(«) J'{^) J [a) ^ 

IS the value oi .,. n • 

If all the differential co-efficients of both terms oC the fraction 
become infinite up to the (n, —■ 1)"' order inelusivolv. thou 

F{a) ^ F^{a) ^ F\a) _ _ F^"\a) . 
/(a) ~ f\a) ~ r\a) " •" - yv<^^^^^ ' 

and the true value oi' a ratio, that, for a particular vahie of tho 
variabU^j takes the form ^'^ , is tlu^ valuo ot'tlio ratio oi' iho dif- 
ferential co-ellioients oi' the order oi' that tirst found, whoihor 



156 



DIFFERENTIAL CALCULUS. 



of numerator or denominator, which does not become infinite 
for the assigned value of the variable. 

Example. For x = 0, 



cosec. X 



sm.- X 



2 sin. X COS. x 



X COS. X COS. X 



X sm. X 



rr:0. 



103, The rules which have been given for finding the true 

OC 

value of ratios which take the form - or — are applicable for 

CO ^^ 

infinite as well as for finite values of the variable. This fol- 
lows from the fact, that the reasoning by which these rules 
were established requires only that the value attributed to x, 
causing the fraction to assume the one or the other of the above 
forms, should be the same in both terms, but does not involve 
any supposition in regard to the magnitude of this value. The 
rule depending on differentiation may be demonstrated directly 
when the form of indetermination comes from the hypothesis 

X zz:ioz . 

Represent the terms of the fraction by F{x), f{x), as before, 
and suppose, that, for a:; z= oc, we have either F(^x) =. 0,f{x) = 0, 

or F(x)= OC, /{x) = Gc : then, putting - for x, 



f(- 






/(^; 



but, b}' rules already given, 



'KJ)^ 




i^Q^ 





1^,-/1 



1 /I 

- f 



— 1 









INDETERMINATE FORMS. 



157 



iience 



^K-)l 




rwi)l 


w 


L — -i 


\y/ 


/i\ 




. /i\ 


f{-) 




^l;;) 


I w. 


2/ = 


L w J 



_/:?!i^ 
~\/'(^) 



2/ = 

Ex. 1. For X r= 00 we have, Avhen a > 1, 

X 1 

Ex. 2. When x := oo , 

\jC octet 

Ex. 3. When a:; = oo , and ti is the integer which immedi- 
ately follows a, 

x^ __ a{a — l){a—2 ) . ..{a —n+1) __ 

104. Form x co . Let F{x),/{x), be two functions of x, 

one of which becomes 0, and the other infinity, for a particular 

value attributed to x. For x == a, suppose F{x) = 0,/(a;) =i oo . 

The product may be put under the forms u = F[x)x/{x) = 

F(x) fix) 

— ^ — -i ? the last two of which, for the assigned value 



/{xf F(x) 

of X, take respectively the forms -, -, and can therefore be 



0' oo 



treated by the preceding rules. 
Ex. 1. 



^c=zli2 — ^]X tan. ' == Oxoo when x = a. 
^ a 2a 



But 



/ 1 2 — - 1 X tail. 



12 



nx 

tan. ^r- 

2a 



cot, 



TTX 

2a 



158 



DIFFERENTIAL CALCULUS. 



and 



12 



TtX 

cot. — - 

2a 



1 




1 


a 


2 


X 

a 


7t 




1 



2a 



Ex. 2. Foro^^zO. 

odx = — X oo , 

, Za? ^ I lx\ 

xlx = — 7, and ( - — - 



sm/ 



,-1 



2a 



:=0. 



Ex. 3. x^ilxY^zn^Xoo for cc 1= 0, when the exponents w 
and n are positive. 

Make x— -^, then x"^ (Lxf — {— If '^ . This, by Ex. 3, 

Art. 103, is zero when ?/ =oo, which answers to a? = 0. 
105. Forms 0^ <>o«, ±1". 

In the exphcit function y= (F{xyj ^ of the variable x^ 
suppose that F{x)^ /(^)j ^'^^ such, that, for the particular value 
x:^a, y assumes any one of the above forms ; then, to deduce 
a rule for the evaluation of y, we proceed thus : — 

Take the Napierian logarithms of both members of the 
equation y= fF(x)J , and we have 

Now, since, to have one of the proposed forms, F{x), for the 
assigned value of x, must take one of the values 0, oo, or 1, 



lF{x) will become either — oo, -f- oo, or 0, and 



lF(x) 



A^) 



will take 



one or the other of the forms - , ^ , and may therefore be 



INDETERMINATE FORMS. 



159 



used for calculating the true value of ly, from which we pass 
to that of the function itself 

Ex. 1. x"" for ic ==: becomes O''. In this case, — ^-^ = — , 

fix) X 



which, for cc = 0, is equal to 

r 1 



1 



= 0: .-. ?yz=0,2/ = l. 



J ar=0 



Ex. 2. xoo = oo'' when ir = oo . 

Here - — \~^ =i — ; and this, when a? = oo , z= - zn : 

1 X ^ x 



/(^) 



.-. ly^^,y = l. 



Ex. 3. a^i-^ = 1" when x^\, 

IFix) Ix , 

hj~—l: ... y:=^e-\ 

106. Form oc —co . 

If the functions F(x), f{x), of x, both become infinite when 
x :=a, then, for this value, 

F{x) ~f(x) = 00 — oo . 

To deduce a rule for the evaluation of expressions that take 

this form, make F{x) — y, ., f{x) — tt-v-tJ then tlio value of 

x that causes i^( a:), /(.r), to become infinite, must rodiice F^[J'), 
fi(x), to zero; and, if a bo this vahio oi^ x, we have 

and the case is thus matk^ io fall under the rule of Art. 101. 



160 DIFFERENTIAL CALCULUS. 



Ex. 1. Sec. X — tan. x=zoo — oo when a; = -, 

, 1 sin. X 1 — sin. x 

sec. ic — tan. ic= i=: 

COS. a; COS. ic cos. x 

T /I — sin. ir\ /cos.ccX 
and TT = ^ TT =0. 



COS. X. 7^=2" \sin. xj'''=r^ 

1 aj 

Ex. 2. 7 ,— =z Qo — oo when x :=r 1, 

Ix Ix ' 

1 x\ (\ — x 



iiX ox I „ __ 1 \ iX J -I 

107. It may happen that not only do F{x), f{x),uii\iQ 

Fix) 
ratio yf-y, vanish for the assigned value of the variable, but 

also all their successive differential co-efficients, however far 
the differentiation be carried. For suppose jP(cc) =a~in, 
which becomes for a; == if a and n are positive, and a > 1 ; 
then 



_ 1 



,n + \ 



F"{x)^nla.a-"Tij^-''-±l 
Making a? = -, these differential co-efficients become 

a' 

It is needless to carry the differentiation further to see that 
each differential co-efficient will contain a factor of the form 

— ^, in which a, m, and n are positive, and a > 1. This factor 
takes the form ^ for 2 = go ; and if we apply to it the method 



INDETERMINATE FORMS. 161 

for finding the true value of such expressions by differentia- 
tion, differentiating p times, p being the whole number next 
above m, z Avill disappear from the numerator of the ratio of 
the differential co-efficients of the order p^ and this ratio 

Ic 
would be of the form — — r , in which ^ is a constant, and wiz) 
9(2) ' ^^ ^ 

a function of %, that becomes infinite when 2 z= 00 . Therefore 

all the differential co-efficients of F{x) vanish when a? = 0, 

_ j_ 
which answers to z ^= 00 . Hence, if /(x) = & ^^, the terms 

of the ratio -—-^ ^ and all their differential co-efficients, vanish 

for ii; = 0, if a, h, n, g, are positive, and a and h are each 
greater than 1. The true value of this ratio cannot then be 
found by the method of differentiation. 

When 71 = g, the ratio becomes [ j •*'', the true value of 

which, for a? = 0, is 0, if a > h, and 00 if a <^ Z>. 

108, The solution of cases of indetermination is often 
facilitated by transforming the example so as to make it take 
a form of indetermination different from that under which it 
presents itself. Thus 

_' \ 

= - becomes Avhen .r = ; 

XX 

but 

e"- 1 I , 

— = 1 := ,, - when X = : 

X a;c'x X 00 

and the true value of tlio given expression is 1 divided hv 

1 
the true value of ar ' Avhen x = 0. 
21 



162 



' -\ 


r 1^ 




xe^ ) = ^ 


T 


r= - 


. A=o 




x = 



DIFFERENTIAL CALCULUS. 

'- 1 1 

e" X — - 



r^^ 



07=0 



'~l]=^=0. 



Again : if F{x) becomes infinite when ic =z oo , then (Art. 
102) 

But (Art. 56) 

and it is evident, that as x increases, and finally becomes infi- 
nite, the second member of this equation converges towards 
and finally becomes F\x) : hence 

'^{^)\ _ /F{x + A) - F{xf 



'^ J X = oo \ '^ 



or by making 7i z= 1, as we may, since h is arbitrary, 



If, now, the value of fF(x)jx, when x ^= oo , is required. 



1 lF(x) 

F(x)f = e ' (1); 



we have 



a true equation, as may be seen by taking the logarithms of 
both members : therefore 



F{x)y =(e 



lF{x) 



(2); 



and the proposition is thus reduced to the evaluation of 

]l^^ lF(x) 

e ^ , or rather of — ^— ^ when a; = oo . 
' X 



INDETERMINATE FORMS. 163 



'-^•L.=(-<'+"--'^'Lr('^^'L.<^'' 



By what is proved above, 
But, from Eq. 1, we have 
therefore, by Eq. 3, 



^<"tf(^^ 



Let this be applied to the determination of the true value of 



- y when cc = oo . 



1.2.,. xj 

Now, by what has just been proved, the required value is 
that of 

1.2.. .(x+1) -^^ = [~r) "^V+x) fo'-^ = -- 
But (Art. 9) fl+-T =<■• 



:r=x> 



109, Thus far, the discussion of the indeterminate forms 
has been confined to functions of a single variable. A few 
cases will now be considered in which these forms present 
themselves in functions of more than one variable. We re- 
mark, that a function of two variables mav assume the form 

0' 

either Avhen a particular value is attributetl to but one of the 
variables, or when both variables have particular values given 
them. An example of the first case is 
h(x — a) 

whicli, for .T— a, reduces to , wliatevor bo iho vahio oC f/ : 



164 DIFFERENTIAL CALCULUS. 

but by dividing out the common factor x — a, and then making 
x^=.a, we have s =: -^ — 

An example of the second is 

c{x —a) 






m{y — h) 



which takes the form - for a:; =a, ^ = &, and, for these values 
of the variables, is really indeterminate. For let p denote the 

ratio , , then 2 =: -^ ; and, since x and y are independent, 

p is an arbitrary quantity, and z is therefore indeterminate. 

HO. To investigate a rule for the evaluation of the inde- 
terminate forms of functions of two or more variables, take the 
function u =: F{x, y\ x and y being independent, and suppose 
the function to be finite and continuous for all values of x and 
y between a; = a, a; = a + Z^; y = ^/2/ — ^ + ^/ ^^^ further, that 
all the partial diiferential co-efficients of the function, up to 
{n — 1)^^ inclusively, vanish for a^ =tt, y ^zh; but that those 
of the n^"^ order neither vanish nor become infinite for these 
values of x and y. 

For the time, denote by ht, ht, the increments of a and b; 
so that the function, when the values of x and y with their 
respective increments are substituted, is F{a + ht, b -\- Jet), 
which becomes F{a -{-h,b-{- h) by making ^ — 1 : then de- 
noting F(a -\- 7it, b + kt), which is a function of t, hy/(t), we 
have 

F{a^M,b-\-U)^f(t) (1); 

and, making in this ;{ 1= 0, 

F(a,b}=fid) (2). 
lf/(t) is finite and continuous for all values of t, from ^ == 



INDETERMINATE FORMS. 165 

up to ^ = any assigned value, t=:t; and if, in addition, all the 
differential co-efficients of f{t), up to the (n — 1)*^ inclusively, 
vanish for ^ = 0, while that of the n^^ order is neither zero nor 
infinite for ^ r= 0; then (Art. 56) 

/(0-/(0)=j-2^/«'(»0 (8). 



To simplify the application of this equation to our purposes, 

ake x^ = a -{- lit, 
f{t) = F{x',y'): 



make ic' = a 4- lit, y^ = 6 + ^^ ; whence -.^ =:/i, -^ = k, and 



_ ^^ dx' dF dy _ dF ,dF 
•^ ^^ ~ d^'di ^'dUf' llt~d^'^d^' 

Making ^ = 0, observing that then x' z=za, y' ^^ h, and that 

d F dF 
what -y-y, y— ^, become, will be identically the same as what 
ccx ^y 

dF dF 

-,- , -,-» become when a^ =a, ?/ =i 5, and denoting' these dif- 
dx' dy' u ^ to 

ferential co-efficients for this value of ^ by ( ~~ ) , ( — ) , we 

\dx /o \dy /o 
have 

If ( — \ A - I J both vanish, then f (0) = 0, and wo must pro- 
\dx }l \dy ), , J \ / , I 

ceed to the 2d differential co-officiont of /[t], Avliich is 
•^ ^*'- dt ~dx'-'' ^^dxUh/ '''■ + ,/,/'-■' • 

4 

and, in this making ^ = 0, wo have, by adopting a notation iu 
harmony with that iu the expression for /'''(^O), 



166 



DIFFERENTIAL CALCULUS. 



and in this also 



become when 



... are what , 

^dx'- /„ dx'^ 

X =z a,y ^=h. If all the partial differential co-efficients of the 

second order vanish for ^ = 0, then /^^(O) = 0; and we must 

pass to the 3d differential co-efficient of /{t), and in this make 

t zzzO. We should thus find 



/-(0) = 



d'F 
dx^ 



A3 4-3 



\dx - dy /o \dxdy - J dy 



and so on; the expression for /^"\t), all up to that vanishing 
for ^ =: Oj being 



d"F 
dx^'' 



h"" -\-n 



d^'F 



dx'^^~'^dy' 



-p-iA;H \-n 



d'^F 



dx'dy"'' 



lik""- 



+ 



d'^F 
dy"" 



k", 



the laws governing the co-efficients and exponents being ob- 
viously the same as in the Binomial Formula. 

Now, since F{x,y), F(x', y^), differ only by having x and y 
in the one replaced respectively by x^ and y^ in the other, it 
follows that any partial differential co-efficient of F{x, y) will 
be the same function of x and y that the corresponding partial 
differential co-efficient of F(^x', y') is of x' and y' ; and hence 
the hypothesis that renders x ^=.x'^ yz=y'^ will, at the same 
time, cause these differential co-efficients to be equal. There- 
fore make x ^^ x' :=. a -\- ht, y z=y^ =^h -\-Jct, and we may 
write 



drF. 



r\t)^\ 



dx" 



/i" 



d-F 



' dx"" 
d^F 



dxdy^'' 



'dy 



. d^F. 



dy 



h^ 



W'. 






INDETERMINATE FORMS. 167 

all of the several orders of partial differential co-efficients of 
F(Xj y)j up to and exclusive of the n^^, vanishing for a; = a, 
y — b, that is, for j5 = in f'{t) ...f"-^^ {t) ; but all of those 
of the n^^ order not vanishing. Then, writing dt for t in 
Eq. 4, and substituting in Eq. 3, we have 
F{a + lit, h + U)- F{a, h) 

n I w V. « — 1 w 1 / ' 



l.2,..n\dx'' ' dx^'-^dy 

d^F ,, , (/"i^ 



+ "" dxdy--' ''^^ + ^ ^ 7^ =« 



and if, in this equation, we make t z=z 1 ] then 

F{a -\-7i^h-]-k) — F(a, h) 

• " + '' dxd.y—'''''^ + %"- /"^^.t ^' ^^'^' 
which enunciates a theorem relating to a function of two inde- 
pendent variables analogous to that demonstrated in Art. ^(^ 
for a function of a single variable. 

In Eq. 5, suppose both a and h to be zero, and then change 
h and h into y and x, as Ave may do, since li and h are not only 
independent of each other, but may have any values, and we 
have 

d"F d"F \ 

which expresses auoihiM* thooroiu n^ladng loa t'uiu'tion ot'iw^^ 
independent variables similar to that iu Art. oC) lor a function 
of a sina,'lo variable. 



168 DIFFERENTIAL CALCULUS. 

111. Let F{x, y), f[x, y), be two functions of the inde- 
pendent variables x and y, and suppose that not only the func- 
tions, but also all of their successive partial diiferential co-effi- 
cients, up to those of the {n—iy^ order inclusively, vanish for 
X ^= a, y :=: b ; but that, in respect to those of the n^^^ order, all 
do not vanish, nor do any of them become infinite for these 
values for x and y: then by Eq. 5, Art. 110, remembering 
that by hypothesis F{a, h) == 0, /(«, h) — 0, we have 
F{a + A; 5 + h) 

1 fd'^F. , d-F ,„ ,, , 



\.±Z.,,n\dx'' ' dx^'-^dy 



d^'F d''F 



y = b-^Qk 






dxdy''-'^ ' dy 



:6+| 



Dividing (1) by (2), member by member, we have 

F{a + 7^ 5 + h) 
f{a + h, h -fk) 

Now, the increments li and h are quite arbitrary, and, like 
the variables to which they refer, are also independent of each 
other : we may therefore assume k = mli, in which m is an ar- 
bitrary constant. 

Substituting this value for k, in Eq. 3, dividing out the 
factor Zi'', common to the numerator and denominator of the 
second member, and then making h =^ 0, we have 



INDETERMINATE FORMS. 169 

F{a^) _ 
/(a,5)-0 



d^F , d^F , , d^F ^ ^ . d-F , 



(4). 






d-^f . d-f . , d-f ^_^,d^f 

t/^" ' dx^'-^dij ' ' tfa:^^" ' £/?/" 

This value of — 7-^, = x is indeterminate, since m is arbi- 

trary ; and generally, if two functions of two independent 
variables both reduce to zero for particular values of the 
variables, the ratio of the functions for such values is really- 
indeterminate. 

112. Making n=^l, in Eq. 4 of the last article, we have 

dF , dF 
F{a, h) __ dx ' dy 
J(^b) - d/^df— ' 
dx dy 

and this value of ,.\ ' ,, becomes determinate if ^— , -^, both 
/ {a,o) dx^ dx 

vanish for a; = a, 2/ ^= ^/ o^; ^^^^7 remaining finite, if -, , y, both 

vanish for these values of x and y. The value of — rV-^ = . 

becomes, in the first case, 

dF ' dF 

Fictj h) dy , . , . Fia.b) dx 

- ) ' — •; ; and, ni the second, } '! — ^ -. 

dy dx 

dF dF 

F(a,h) .11, • . T ^^•^' ^^!f 1 1 wi 1 

,.-) '^x- IS also detornunate 11 , , = , . : wo slioukl tIumi luivo 

dx dy 

dF/df df \ (fF dF 
F{a, b) ^ dv. \dx ]^ dy'y _d7v_ dy 
'/{a, b) ~ df /df df ^\^ - df ~ df 

dx\dA' i/y dx dy 



170 DIFFERENTIAL CALCULUS. 

If J- = 0, ^ — 0, -i- — ^, f — 0, we make n~2\n Eq. 4, 

Art 111, and thus have 

d'^F , ^ cr^F , c^-^i^ 2 
F(^a,h) _ dx' ^ dxd y ^ d y^ . 

dx'^^ dxdy d.y^ 

which is indeterminate, except in particular cases depending 
on the absolute and relative values assumed by the partial 
differential co-efficients for the values x^=.a^y ^^h. 

Example, z == — , 7" "^ , == -- when ic = 1, ?/ = 1. 

Here F{x, y) — lx-^ ly, f{x, y)z=x-\-2y — 3, 

^=^:=1 f0rx:.= l- ^^=1 

dx X ^ ^ dx ^ 

dF 1 ^ . A df ^ , 

-,^ z= == 1 tor v = 1 ; 7- = 2 ; hence 
dy y ' ^ ' dy 

— 1.+ ^ • 

and therefore, for the assigned values of x and y, the function 
is really indeterminate, and may take any value between -|- co 

and — oo . 

118» In the case of the implicit function u = F{x, y) == 0, 
we have found 

dF 

dy dx 

'dx~~l¥ (^)- 
dy 
Now, if oj := a, y =r &, are values of x and ?/, which, while they 

satisfy the given equation, at the same time make -^ = 0, 



INDETERMINATE FORMS. 171 

-, =r 0, then -^ takes the indeterminate form - , and its true 
ay ax 

value, if determinate, must be found by the preceding method. 
Differentiating numerator and denominator, we have for 
x—a,ijz=:h, 

dF d}F d'F dy 

^y dx dx'^ dxdy dx /2\ 

dx " ~dF'^~ 'dW~~WYdy ^ 
dy dxdy dy'- dx 

from which we get 

dy'^ \dxj dxdy dx dx^ 

dn 
a quadratic with respect to Jr This equation agrees with 

ax 

dF 
Eq. 3, Art. 96, observing that by supposition =0. It must 

be remembered that Eqs. 2 and 3 are true only for the partic- 
ular values X = a, ?/ = Z>. When these values of x and ^, in 
addition to making the function and its first partial differential 
co-efficients equal to 0, also make 

(^F __ d''F _^ ^'^-0 
dx^ ' dxdy ' dy'^ ' 

the value of ' ' , as p'iven b\^ Eq. 2, ao-ain takes the form : 
dx^ ^ ^17 ^. ^) 

and Avo must in that case effect a third differentiation, which 
gives 

d'F 2 ^^^F dy (PF_ /dyVi d-F d'-y 
dy _ dx^ dxUiydx dxihr\dx/~^'d,v(fydx- 

dx ~ ~ d'F~~~^:^d'F~dJ~~d\F /dy\^^ d'F a-y ^ ' 

dx-dy "" dxdy'^ dx dy' \dx/ dy^ dx^ 

and from this, obsiM-NiiiiT that b\- h\noihcsis , , in: 0, , ,=:0, 

' • dxay ■ dy'^ 

we derive the cubic cipiation, 



172 DIFFERENTIAL CALCULUS. 

dJF/dy^^ 3 i^ W+ 3 --— ^ + — = (5). 
dy'^ \dx) dxdy'^\dx) dx^dy dx dx^ 

diJ 
Ex. 1. Determine the value of -^ from ax'^ —y^ — hy'^=z 

dx 

when a? = 0, ?/ = 0. 

Here -^ = —— — - — - lor ic = 0, v = 0. 

But, for these values of x and ^Z, we have 

dy _ 2aaj _ 2a 2a _ 

5a.-3F+%-rW7^~27^ tor.._0,y_0: 
dx dx dx 

dx'' -^djj^ \dx) ~b^ dx~ 
dx 
Ex. 2. If ?^ = ^'^ + 3a2^2_4^2^^_^2^2^0, find the 

value of — for ir = 0, ^ = 0. We have 
dx 



dy __ a /dy\i^a dy __ 



la 



du 
dx 

-^ := — 4:a^x — 2a'^y: 
dy ^ 



z= 4:X^ -{- Qa'^x — ia^y. 



dy _ 4:X^^6a'^x — 4:a^y __ 2x^ -\-^a^x — 2a'^y 
dx 4:a^x-\-2d^y 2a^x-\-a'^y 

— -ioYX=z{)^y = 0. 

Differentiating both numerator and denominator with respect 
to X and y, we get 

6^^- + Za' - 2a' ^^ Sa^ _ 2a2 ^V 

dy dx dx 

-Y- = 7 = 7- tor cc = 0, y = 0, 

2a^ -\- a^ -^"^ 2a' -\- a' ~ 

dx dx 

3 - 2 -t 
ax 



^+2'- 



INDETERMINATE FORMS. 173 

dx \dx) dx^ 

Should it happen that the particular values of x and y reduce 

d'^^F d'^F d'^-'F 

-——- to zero, while — — — , -— — , remain finite for these values, 

dy^ dxdy dx- 

dti 
then Eq. 3 of this article gives, for one of the values of t~' 

ctx 

d'^F 

dy _ dx'^ 

dx~~ d'"F 

A 

dxdy 

dii 
which is finite, while the other value of - becomes infinite, 

cix 

as may be shown by discussing the equation ax^-\- hx -|- c=:0, 
under the suppositions that a = 0, and that b and c are finite. 

114, The investigation of the true value of - , when it 

dx 

takes the form - for a; = 0, y =: 0, may be simplified by the 



consideration, that, in this case, ( — L^^ = f ' ],.^^) ? as is evi- 

dent from the definition of differential co-efficients. Take Ex. 
2 of the preceding article, and divide through by x'- ; then 

^2 + 3a'' - 4a'^ '[ - a' ('^\ = 0. 

Solving this equation with roforoncu^ to ' , and thon making 
X = 0, we find, as before, 

ax X 



174 DIFFERENTIAL CALCULUS. 

In like manner, the example 

x^ -\- ay^ — ^axy"^- — Zax'^y = 0, 
which gives 

dy_ 4.x' - 6axy - 2ay ' _0 n r. r. 

dx-2>ax''^4.axy-^2>ay'-{) ^^^ ^ - ^^ 2/ - ^, 

by dividing through by x', takes the form 



sXJ \XJ X 

a cubic equation, from which, after making a; = 0, we get for 

V 

- the three values 0, 3, and — 1. 

X 

For another example, take the equation 

x^ -\- ax'^y + hxy'^ — ?/^ = ; 

whence a^ + a- + 6 ( - j — ?/ (- J ziz 0, 

X \ / \ / 

which, for ic = 0, ?/ = 0, reduces to 



X \x 

It If Oj 

and therefore we have - = 0, and - = • 

X x h 

By dividing through by y', the assumed equation becomes 
/x\ ' /x\ '^ X 

which is satisfied by making simultaneously cc = 0, y = 0, 

/v» XI] 

- = : hence - = 0, or - = oo , will satisf>^ the given equation 
y y X 

in connection with the values x = 0, ?/ := 0. Therefore, when 

y 
X and y have these values, - may have the three values, 

0,--, ». 



INDETERMINATE FORMS. 175 



EXAMPLES. 



U"-iA=. 


Ans. -♦ 
n 


/x — sin."~^fl:\ 


Ans. - ^. 


V sin.^a; y^^o 


'e^ — 2 sin. a^ — <?~A 


Ans. 4. 


ic-sin.aj A^o 




Ans. 0. 



1. 

2. 
3. 
4. 



In solving this example, begin by making ^^ __ _ . ^-^ence 

i_ 
2 i:= 00 when ir = ; and we conclude that ^~^' decreases 

more rapidly as x decreases than does ic^", however great be 

the value of n. 

1 — X -\- Ix . 

Ans. — 1. 



1 — V 2a; — x-j^ ^ 1 

nx , nx , nx , , vx\ — 

a, +a^ +a, + • • • +«. V Ans.a,rt.,a3...a,. 

^^-l+ i£^) Ans. 0. 

8. ( 2^ sin. ^.j Ans. a. 



\1 — cos. 7lX/j._o 



2 

Ans. — ; 

n- 



10. (l+-y Ans. 1. 



SECTION XL 

DETERMINATION OF THE MAXIMA AND MINIMA VALUES OF FUNC- 
TIONS OF ONE VARIABLE. 

115, When the value of a function, for particular values 
of the variables, is greater than those given by values of the 
variables immediately preceding or following such particular 
values, the function is said to be a maximum : when it is less, 
it is a minimum. To fix attention, suppose y z=.f(^x) to be 
such a function of x^ that, as x gradually changes from a spe- 
cific value to another, y undergoes continuous changes; but, 
having increased up to a certain value, then begins to de- 
crease, or, having decreased to a certain value, then begins to 
increase. The value of y at the point where, from increasing, 
it begins to decrease, is a maximum- ; and at the point where, 
from decreasing, it begins to increase, it is a minimum. In 
the first case, the value of y is greater, and in the second case 
less, than those which immediately precede and follow. The 
terms maximum and minimum must be understood as relative 
rather than absolute ] for it is plain that a function may have 
several maxima and minima as above defined. 

Confining ourselves, for the present, to explicit functions of 
a single variable, we have seen (Art. 52) that such function 
can pass from increasing to decreasing, or the reverse, only 
when the first differential co-efiScient of the function passes 
through or oo , or when this difi'erential co-efficient changes 
from positive to negative, or from negative to positive. 

176 



MAXIMA AND MINIMA. 177 

Hence those values of x which render y z=f(^x) a maximum 
or a minimum must be found among those which satisfy the 
equations /'(a?) =: 0, f'{x) = oo . 

Let xi=a be a root of one of these equations, and let h be a 
very small quantity ; then/(o!) will be a maximum \ff'{a — h) 
is positive, and f'{a-\-h) is negative ; but /(a) will be a mini- 
mum \i f [a — h) is negative, and/' (a 4-^) is positive. When 
f'{a — h), f [a -j- /i), are both of the same sign, whether posi- 
tive or negative, /(a) is neither a maximum nor a minimum. 
Ex. 1. y —f(^x) = 2ax — x^, 

f {x) = a — X ; f'{x) = gives x^= a, 

f^[a — h) =z a — a-\-h^= -\- h, 

f [a -\- h) z=z a — a — A = — 7i. 
Hence cc = a renders the expression 2ax — o:^ a maximum, as 
may be easily verified ; for, making cc = a, 2ax — x''- reduces 
to «-; but making ^r = a -J- 7^, or x ^^a — h, our result in 
either case is a^ — A^ < a^ 

116, The method just given for deciding whether or not 
a root of the equations f'{x) = 0, f'{x) = oo , answers to a 
maximum or minimum state of /(a:), is general ; but, in respect 
to the roots of the equation /'(j::) = 0, we may for this pur- 
pose deduce a rule, that, in many cases, admits of easier appli- 
cation. 

As before, let x=.a be a root of /'(^•) = 0, and suppose 
that f^"\x) is the first among the derivatives of/(x') that does 
not vanish for this value of .i-; then (Art. oG) 

f{a + 70 -J\a) = ^l'^^^ f"'\a + Oh ^ ^\\ 

Since is a proper fraction, anil /^ as we shall suppose it to 
be, is a very small quantity, it is obvious that the sign of 
/''"\a -{- Oh) cannot change Avith that of 7^ and is therefore 

23 



178 DIFFERENTIAL CALCULUS. 

invariable : hence the sign of the second member of Eq. 1 de- 
pends on that of h"^ when combined with that of /^"^(a + dh). 
But, if n is an even number, the sign of li"" is positive, what- 
ever be the sign of li ; and in this case the sign of the second 
member of Eq. 1, and consequently that of /(a + h) —f(a), 
will be the same as that of f^"\a -\- 67i), or as that of /^"^(a), 
since f^^^^a -\- Oli) and/^"^(a) have the same sign. If, then, n 
being even, f^''\a) is positive, /(a + ^0 ~f{^) ^^ ^^^ positive, 
whether li be positive or negative, which requires that /(a) be 
less than /(a ± h) ; that is, f{x)^^^ ^=^f{<^) is less than those 
values of/(::c) which are in the immediate vicinity of this par- 
ticular value. This condition indicates a minimum state of 
the function. But, n being still an even number, if /^"^(a) is 
negative, then /(a dz h) — /(«) is negative, which requires 
that /(a zb li) be less than /(a) ; and a maximum state of the 
function is indicated. 

The hypothesis in respect to 1i being continued, if n be an 
odd number, then, since ( -|- hy and ( — hy have opposite signs, 
and the sign of /^(a -{- Oh) does not change with that of /i, the 
second member of Eq. 1 will change its sign as li changes from 
positive to negative, or the reverse. Hence /(a -\- h) — f{a) 
and f{a — h) — f{a) must have opposite signs, and f{a) is 
greater than one of the expressions f{a -]- 7i), f[a — A), and 
less than the other; that is, f[a) is neither greater than both 
the immediately preceding and immediately following values 
of the function, nor less than both these values; and therefore, 
in this case, x:=z a renders the functiofl neither a maximum 
nor a minimum. Whence the rule for deciding which of the 
roots of /'(x) := corresponds to maxima or to minima of /(x). 

" Substitute the root under consideration, in the successive 
derivatives of the function, until one is found that does not 



MAXIMA AND MINIMA. 179 

vanish. If this derivative is of an even order, and the result 
of the substitution is positive, the root will render the function 
a minimum; but, if the result of the substitution is negative, 
the root will correspond to a maximum. If the first of the 
derivatives of the given function that does not vanish is of an 
odd order, the root corresponds to neither a maximum nor to a 
minimum state of the function.'' 

Ex. 1. Find the values of x that will render 
/{x) — x^^- Ox'' -i- 24:x — 7 
a maximum or a minimum. 

f (x) = 3x2 _ ij^^ _|_ 24 = 3(^2 — 6^ + 8), 
/-(x) = 6(x-3). 
From f^{x) = ^(x'^ — 6x -{- 8) =: 0^ we find a; = 2, or a; = 4. 
When x = 2, f"{x) = 6(x - 3) =r — 6 ; and hence, for x = 2, 
the function is a maximum. When x = 4:, f'^{x) = -[- 6 ; and 
ic = 4 renders the function a minimum. 

117, When y is an implicit function of x given by the 
equation F(x^ y) z=z 0, and the values of x corresponding to 
the maxima or minima values of y are required, we may, in 
cases in which the resolution of the equation with respect to 
y is possible, employ the methods of the preceding articles. 
But, without solving the given equation, wo may proceed as 
follows : — 

Let u = F{x, y) = 0] 
then (Art. 84) 

die 
dy _ dx 
dx ~ da. 

Limiting our discussion to (lie values o^ x derived \\on\ the 

(/// ,, ... du . ,. . di/ ^, . , du 

equation ,' -^0, il , is liuite, ; — roquires thai , — 0. 
dx di/ dx dx 



180 DIFFERENTIAL CALCULUS. 



Hence we have the two equations 

du 
dx 



u — ^, -rr_ — ^', 



by the combination of which, eliminating y, we get a single 
equation, in terms of x, which will determine those values of x 
which may or may not render ?/ a maximum or minimum. 

d" u . . du 
To decide this, we must pass to -^-y, which, since 3- = 0, re- 
duces to 

d'^u 
d'^y __ dx"- . 



dx"- du 

dy 

and if the values of ic and y derived from the equations i^ = 0, 

du 

-— = 0, do not cause this to vanish, but make it positive, the 

value of y corresponding to this value cf cc is a minimum ; but 

d}y 
if, by these substitutions for x and y^ -y-| becomes negative, 

(JjX 

the value of 2/ is a maximum. But, if these values of x and y 

/72 nj d 11 

cause -7—^ to vanish, -^-~ must also vanish in order that y 

may be a maximum or minimum ; and it would be necessary to 

d^y 
find -j\ , and substitute in it, to enable us to decide whether 

2/ is a maximum or a minimum. 

Ex. 1. Find the value of x that will render y a maximum or 
minimum in the function 

u =i x"^ — Zaxy -\- y^ z=. ^ (1). 

du 

dy dx __ay ^ x^ . 

dx~ du y'^ — ax 

dy 



MAXIMA AND MINIMA. 181 



-^ zz: answers to ai/ — x"^ z=Q (2). 

From (1) and (2), we find 

x^ — 2a^x^ = 0: 
therefore cz: = 0, or i;c = a\/2, and the corresponding values 
of y are y ^= 0, ?/ = a-y^^. 

The values x z=z 0, y = 0, in the expression for -' - , cause it 

to take the form -; and the true value of ~- must be found. 
ax 

This maj be done by the method of Art. 113. It is better, 
however, to proceed as follows : — 

The second and third derived equations of the given equa- 
tion are 

<^— )S+(«''l--)S'+KSJ+--;^ 

and when, in these, we make ic i= 0, y = 0, we find — = 0, 

ctx 

d'^y 2 
and — — = -— • Hence ?/ = is a minimum when ^' = and 
ax^ 6a 

y ^0. When x = a^2, the corresponding value of y^=a y 4 

is a maximum ; for Ave have, in this case, 

d'l ^ _ dx^ _ _ _2f_ ^ 2a ^J2 ^_2 

dx^ dio y'^ — ax rt-.y/lG — a- -^2 a' 

which indicates a maxinuim. 

lis. Suppose that the relation between x, //, and u, is ox- 
pressed by the twt) et|uati(>ns u = l'\x, y), j\a\ i/^ - = 0. so 
that'i^ is implicitly a ruuction of .r ; for, doduciiiLL- the value ot" i/ 



182 DIFFERENTIAL CALCULUS. 

in terms of x from /(x, t/) = 0, and substituting this value of 
y mu^=^ F{x, y), we should have ic an explicit function of x. 
If the maxima and minima values of to were required, we 
might pursue this course ; but we may accomphsh our purpose 
without solving the equation /(a:, y) — 0. 
We have (Art. 82) 



also (Art. 84) 



du 


dF 


dF dy 


dx~ 


~ dx 


dy dx 
df 


dy 




dx 


dx 




df • 
dy 

df 


du 


dF 


dFdx 


dx- 


dx 


'du'df 



hence 

ax dx ay 

~dy 

Xow, the values of x and y which satisfy simultaneously the 

equations f{x, y) = 0, and =0, or the equivalent of the 

ax 

latter, 

dJF df_ _ dF df_ 

dx dy dy dx ' 

but which do not cause — — - to vanish, will render u a maxi- 

dx- 

d^ii 
mum or a minimum, according to the sign of -y^' But, if 

d'^'U 

-r— ^ vanishes for these values of x and y, we must, as before 

explained, pass on to the derivatives of the higher orders, to 

enable us to decide the question. 
Ex. 1. Given 

u = x--^y^-^F(,x,y) (1), 

(^ - «)' + (2/ - Z-)^ - c^ = =/(x, y) (2), 



MAXIMA AND MINIMA. 183 

to find the values of x and y that will make u a maximum or 
minimum. We find 

and therefore 

dFdf dFdf __ 
dx dy dy dx 
becomes 

^{y — 'b)—y{x-a) =0: 

whence ay = l>., y = K. 

ct 

Substituting this value of ym (2), we have 

^2 _ 2a^ _L ^2 -]- _ cc2 - 2 - a^ 4- 62 _ c2 =3 0, 

a^ a ^ 

or ^2^i_|_g^_2^(^a_|_^J^_|-a2 + ^2_,2^0: 

whence 

ac 



Va2 + Z>2 

By differentiation, we get from Eqs. 1 and 2 
d'u _ . 26c2 

Observing that the upper sign in the value of x answers \o 

d'hi 
the upper sign in the value of , ^, and the lower sign in the 

one to the lower sign in the other, avc discover that 

ac 

X=: a -\- 



dx' 



d ij 
renders -^--^ negative; hence this vahic o( x makes }( a maxi- 



mum: but, Avium a , is substituted lor x, \ 



184 



DIFFERENTIAL CALCULUS. 



comes positive, and u is tlierefore a minimum for this value 
of X. 

119. When we have n variables connected by n — 1 
equations, by processes of elimination, we may reduce the 
n — 1 equations to a single equation involving but two of the 
variables, and thus bring the investigation of the maxima and 
minima of the variable which is taken as dependent to the 
case treated in Art. 117. But, in general, it will be found 
easier to operate as follows : — 

Suppose that the four variables, x, y, z, u, are connected by 
the three equations, 

/i(x, y, 2, u) =3 0, f^{x, y, 2, u) = 0, f,{x, y,z,u) = 0; 
and that the maximum or minimum value of u is required, x 
being the independent variable. Differentiating with respect 
to X, we have 



^ "^ dy dx "^^ ^^"^ 



(!)• 



dx dy dx^ dz dx ^^ du dx 
d/2 d/^ dy df2 dz df,^ du 
dx dy dx dz dx du dx 

df^ ,%^y_, % c?2 I % ^^ 
dx dy dx~^ dz dx du dx 

One of the conditions for a maximum or minimum for u being 



du 



, - = 0, introducing this in Eqs. 1, they become 



dx 

dx^ 



+ ;^;7^ + ^- =0 



dfj dy 
dy dx 

d/o dij 

dy dx 



df^ dz 
dz dx 



dfo dz 



dfz , df^ dy 



dz dx 
df^ dz 



dx ' dy dx ' dz dx 



r=0 



(2). 



The equation which results from the elimination of 



dy dz 
dx' dx^ 



MAXIMA AND MINIMA. 185 

from Eqs. 2, together with the three given equations , which we 
will denote by /^ =: 0, /a = 0, /a = 0, will determine values of 
X, y, z, and u. To decide whether any or all of these values, 
or rather systems of values, make u a maximum or minimum, 

(j u 

we must ordinarily pass to -y— 2; and find what sign it takes 

when the values of the variables are put in it. By differen- 

cPu 
tiating Eqs. 1, the resulting equations and Eqs. 1 will give -^. 

120. Before concluding the subject of the maxima and 
minima of implicit functions, we will briefly refer to the limi- 
tations made at the beginning of Art. 117. Resuming the 

equation 

du 

dy dx 

dx du ' 

dy 

we remark, that the necessary condition for a maximum or 

dy 
minimum value of y is, that — change its sign, which it can 

ctx 

do only when it passes through the values or 00 . Xow, 

-f- becomes zero when — - = 0, - being imite ; or when 
dx dx dy 

du du ^ . _ . . . df/ ... 

— = GO , -— benig Iniitc. Agani : ~- becomes nilinito when 

dy dx dx 

du ^ du . . ^ . , d(( di( 

-—1= 0, -- remainnignnite; or wlion — r=: 00 , -- Ihmiil;- linite. 

dy dx dx dy 

It therefore api)oars that the methods heretoforo given lor 

determining the maxima and minima oi' implicit functions an^ 

quite incomplete, as they omit the disoussion o( sovoral oases 

that may give rise to these states ol^ value. 

^lost of the functions Avith whioh we have to deal are those 

24 



186 DIFFERENTIAL CALCULUS. 

whose maxima and minima are indicated by a change in the 
sign of the first derivative when it passes through zero. It 
often happens that the conditions of the problem to be inves- 
tigated enable us to decide some of the questions relating to 
maxima and minima, which, if referred to general rules, would 
require great labor. 

EXAMPLES. 



1 ,, _!,— ^ + ^^ ( When iT =:-, 1^ is a 



2. 


X 


\^ jaiiiiiiiiuiLi. 

("When ic = 1, 2^ is a max. 


^-l + o^-^- 


( ^^ x^= — 1, 'i^ is a min. 


3. 


^^^e^-f 2cos. ic + g--^. 


VV hen cc = 0, i^ = 4, a max. 


4. 


U^ X ^. 


A max. when a? =r e. 



5. Divide the number a into two parts, such that the pro- 
duct of the m^^ power of one part and the n*^' power of the 
other part shall be a maximum. 

rpi , na , ma , 

i he parts are ■ , — and , — : and 

m -\- n m-\-n 



a 



m-\-n 



their product, m"n" [ j when 

Ans. < \,n-\-n/ 

m and n are even numbers. The prod- 
uct may also have two mimimum 
states. 

6. Find a number such, that, when divided by its Napierian 
logarithm, the quotient shall be a minimum. 



X 

The function to be operated with is j- 

IX 



Ans. x=:e. 



MAXIMA AND MINIMA. 187 



7t 



7. z^ = sin. cc(l -|- COS. x). A max. when a; = -• 

o 

nr 

S. u= z — ■ A max. when x = cos. x. 

1 -\- X tan. X 

9. Find the number of equal parts into which a given 
number a must be divided, that the continued product of these 
parts may be a maximum. 

i Each part must be e, the number of 

/ parts -, and the product [e)^ . 

10. Of all the triangles standing on a given base, and hav- 
ing equal perimeters, which has the greatest area? 

Denote the base by h, and the perimeter by 2j9, and one of 
the two unknown sides by x. 

__2p — h (The triangle 
2 I is isosceles. 

11. Of all the squares inscribed in a given square, which 
is the least ? 

Ans. That having the vertices of its angles at the middle 
of the sides of the given square. 

12. Inscribe the greatest rectangle in a given semi-ellipse. 
Let the equation of the ellipse be 

The sides GaJI, . ., and its 

Ans. -I ^ ' -^2 

( area is ah. 

13. Given the whole surface of a cyUnder, rotpiirod its 
form when its volume is a maxiuiuin. 

Represent the whole surface by 2.T(if-. 



1> T 1- ,1 1 ^^ .2(7 

I Kailius iW tlio base . axis 



Ans 



vohinio 



SECTION XII. 

EXPANSION OF FUNCTIONS OF TWO OR MORE INDEPENDENT VARI- 
ABLES, AND INVESTIGATION OF THE MAXIMA AND MINIMA OF 
SUCH FUNCTIONS. 

121, Let it be required to develop, and arrange according 
to the ascending powers of A and h, the function F{x -\-h,y-{- k), 
when F^x, y), and all its partial derivatives up to those of the 
n^^ order inclusive, are finite and continuous for all values of x 
and y included between the values x and x -\-h, y and y -\-k ; 
h and h themselves being finite. 

For the time, replace h and h by lit and kt respectively ; so 
that, when it is desired, we may pass back to li and h by mak- 
ing ^ = 1. Then F{x -\-li, y -\- h) becomes F[x -{- ht, y -\- kt). 
Now, by hypothesis, x, y, h, k, and t, are all independent of 
each other ; and, considered with reference to t alone, we may 

write 

F{x + M,y + ]c{)^/{t) (1), 

Fix,y)=/{0) (2). 

For all values of t between the limits ^ = and ^^ = 1, it is 
evident that /[t) and its derivatives, up to those of the n^^^ or- 
der inclusive, satisfy the conditions above imposed on F{x, y) 
and its derivatives. 

Hence, for such values, we have, by Maclaurin's Theorem, 

/(O =/iO) +/'(0) \ +/"(0) i4 + • • • 

••■+/'°'"W l.2..."("^-l) +/'"'Wl-2n-. (3)- 

188 



EXPANSION OF FUNCTIONS. 189 

Deducing the values of /(O), /^(O), /'(O) . . ./"HO; 1^7 the 
method pursued in Art. 110, except that now x and y are not 
replaced by the particular values a and 6, and substituting 
them and that of/{t) in Eq. 3, we have 

t (dF , dF , 

F{x + Id, y + kt)^ F{x, y)-^^ {^^^-^ h + ^^ k 

^1.2. V/x'^ ^ c?xc/y dy' 

+ ■ ■ 

The notations x z=^x -\- dht, ?/ = y -\- 6kf, attached to the pa- 
renthesis of the last term, signify that in the derivatives 
d'^F d^'F d'^F 

d^^^ d^^dy ^"' dr'"""^ '"^^^'"'^ ^^' "" ^ '^^^^' """'^ ^ ^^ 
y + ^^^if. 

In (4), make ^ = 1, and we have 

dF ^ . dF 
d7j 

1 . 2 y Ix - c/x'(/y ( ly - 

1 A^'^F rP/' d'^F d^F 



(I r (ir' 



+ 



1 /d»F, d"F , ,, (?»F \ 

which is the development sought. 



190 DIFFERENTIAL CALCULUS. 

If, in Eq. 5, we make cc = and y = 0, and then in the re- 
sult write X for h, and y for h, we find 




y = ; 



2/ = 61/ 



^WI^LA (6); 



which is the formula for the development of a function of two 
independent variables into a series arranged according to the 
ascending powers of the variables. 

The extension of formulas (5) and (6) of this article to func- 
tions of more than two variables is easily made. For the 
expansion of F{x -{-Ji, y -{-h, z -\- i . . .), we should find 
F{x-\-7i, y-\-k, z + i...):=:F(x,y,z...), 

dF dF dF. 

dx dy dz 

1 /d'-F^^ d'-F^^ d'F^^ 

1 /d^F d^F ^ d^F ,_ 

+ • • • 

1 (d"F d"F 

'^ \:%7:ti yi^" '*°+rfp *" + ••• 



EXPANSION OF FUNCTIONS. 191 

and if, in this, we first make x,y,z . . .^ severally equal to zero, 
and then in the result write x, y, z . . ., for h, h, i . . ., respec- 
tively, we have 

. ^. . fdF\ /dF\ /dF\ 

+ 1:2:3 jMf+-+(5?^;f^+- 

/ d'F \ ) 



\dxdyy^ 




fd"F\ /d^F\ 



d"F 



dx'^~^dy 



^"■"VH y (8): 



y = On 



a formula for the development of a function of anv number of 
independent variables, and in which the notation ( )^, sio-nifies 
that the variables entering tlie expression within tlio jKiriMi- 
theses are made zero. In formulas (Vom ( ')) to (8) inc^lusivo, A' 
this article, the last terms are remainders expressim;- the dif- 
ference between the value of tlie sum td' the preceding terms 
of the development and (he vahu^ ol' {\\o ("unction. AVhen the 
form of the function under consiileration. and (ho vahics at- 
tributed to the variables, are such, that, as }i increases without 



192 DIFFERENTIAL CALCULUS. 

limitj the remainder decreases without limit, then, by taking n 
sufficiently great, the remainders may be neglected. 

122, Maxima and minima of functions of two or more in- 
dependent variables. 

A function F^x^y, z . . .) of several independent variables 
is a maximum, when, being real, it is, for certain values of 
the variables, greater than F{x + 7i, y -\-h, z -{- i. . .); the 
increments h, h, i . . ., being very small, and taken with all pos- 
sible combinations of signs. On the contrary, the function is 
a minimum, when, under the same conditions, it is less than 
F{x -j- Ji, y -\-h, z -\- i , . .). Let us consider first the func- 
tion F{x, y) of the two independent variables x and y, and 
endeavor to deduce from the conditions of these definitions, 
the criteria of a maximum or minimum of tliis function. 
Resuming Eq. 5, Art. 121, stopping in the second member at 
those terms which involve the third order of the partial deriv- 
atives of the function, and transposing -F(x, y) to the first 
member, we have 

1 /d'-F cV-F , d^F , 



\.2\dx''- dxdy dy 

1 /d'F d'F d'F d'^F \ (1), 

the last term of which we will denote by R. 

Now, if F{x,y) is a maximum, the first member of (1) is 
negative ; and therefore its second member must be negative 
also, and this whether h and k be both positive or both nega- 
tive, or either one be positive and the other negative ; and 
whatever be the values of A and k, provided only that they be 
very smalL If F{x, y) is a minimum, the second member of 



MAXIMA AND MINIMA. 193 

(1) must be positive under the same conditions and limitations 
in respect to the signs and values of h and k. 

But, when h and Jc are taken sufficiently small, the sign 
of the second member of (1) will be the same as that of 

-^/i-f" 77~ ^j which must therefore be permanent and nega- 
tive if F(x, y) is a maximum, and permanent and positive if 
jP(aj, y) is a minimum. It is plain, however, that the sign of 

-^— A -j- ~i-k will change by changing the signs of li and Iz. 

To make the sign of the second member of (1) invariable, 
whether positive or negative, we must have 

and, since h and h are entirely independent of each other, this 

requires that 

dF ^ ,dF ^ ,^, 
^^Cand^^O (2). 

Let X := a, y ^= h, he values of x and y derived from these 

equations, and denote by A, B, C, B„ what ^^, ^^, -^, B, 

respectively become when these values of x and y are sub- 
stituted in (1) ; then (1) becomes 

When the values of 7i and Jc are very small, and only such 
values are admissible, the sign of the second momber of {^) 
will be the same as that of 

Ah' + 2Bhk + a•^ 

which may be put under the form 
26 



194 DIFFERENTIAL CALCULUS. 

The sign of this will be invariable, and the same as that of ^, 
if the roots of the equation 

F + ^ZS + Z-^ W 

are imaginary; ^- being treated as the unknown quantity. 

A/ 

Solving this equation, we find 

Ji _ -B^y/B' -AG 
h~ A ' 

from which we conclude that the conditions for imaginary 
roots are, that A and G have the same sign, and that the prod- 
uct AC\)Q greater than ^^ 

In recapitulation, we say, that ii xz=: a, y = h, make F[x, y) 
a maximum then for these values of x and y we must have 

dF_ dF_ 
dx~ ^ dy ' 

d'-F d'^F , , 
^,^,, both negative, 

d'^Fd'F UPFV 
dx^ dy^ \dxdyj ' 

If x = a, y ^=h, make F{x, y) a minimum, the conditions are 

d'-F d'F 
dy'' 

The existence of real roots for Eq. 4 indicates that we may 
give such signs and values to li and k as to cause the expres- 
sion Ali'^ -\- 2BJik -\- Ck- to vanish, and, in so doing, change its 
sign, which is incompatible with the existence of a maximum 
or a minimum state of F{x, y). 

123, There remains to be examined the case in which 
AG — 5^ = 0. When this condition presents itself, there may 
also be a maximum or minimum value of the function. 



the same, except that then -y— gj 77-2? ^^^st both be positive. 



MAXIMA AND MINIMA. 195 

By the theory of the composition of equations, or by in- 
spection, the expression ^A^ -\- 2B7ik + Ch"^ may be written 

hence, when AC — J?^ =: 0, this becomes —I A- -{-B]^ the 



sign of which, except when [Aj---\-B] vanishes, is the same 



values of -7—;-, j .^ , , -j — 7—^, ~7~r> ^respectively, Avhen x^rza, 



k 

as that of A; and F[a^ h) is a maximum if yi is a negative, 

/ Ji \2 

and a minimum if A is positive. Should iAj-\-B\ vanish, 

7, 7^ 

as it does when - ^ — — , we cannot tell, without further 
h A ' 

inquiry, that F{a-\-Ji, b -\-k) — F{a, h) does not undergo a 
change of sign. To decide this, let L, 31, iV, P, represent the 

d'F d'F d'F d'F 

dx'^ ' dx'^dy^ dxdy'-^ dy 
y z=h ; and also put B^ for the value of 

d^F d}F d^F d^F 

for the same values of x and y. 

Introducing these values, and tlio conditions 

dF dF 

^■^ z= 0, 'J^- = 0, Ah' + 2 r>hh + a- = 0, 

the latter being a consequence of the hypotliesos = — 
and AC — B' =zOj we have 

F{a + h, h + /.•) - F^a, h) :=z 



as that of ^2- S^t ^'^^ have shown, that when A y-\- ^ was 



196 DIFFERENTIAL CALCULUS. 

From this we see, that since the sign of F{a -{- h, h -]- k) — 
I^{u, h), when h and h are very small, is the same as that of 

F{a, b) cannot be a maximum or minimum, unless, if ^ i + ^ 

fc 

vanishes, 

L7i' -\- 33Ih''k + SNJik' + Fh' 

vanishes simultaneously. Suppose these conditions to be sat- 
isfied, then the sign of F{a -\-h,h-\- h) — F{a, h) is the same 

h 
h 

not equal to zero, and li and h were taken sufficiently small, 
the sign of F{a -\- h, h -]-h) —F{a, h) is the same as that of ^; 
but, when F{a, h) is a maximum or minimum, the sign of 
F{a -\- h, h -{- k) — F{a, h) must be invariable for all values 
of li and k which are small enough to cause F{x, y) to change 
in value by continuous degrees in the immediate vicinity of 

the value F[a, h). Hence it follows, that, when the values 

li 7? 

of h and k are such as to make - = — — , these values must 

k A 

give R^ a sign the same as that of A. If these several condi- 
tions are satisfied, the function is a maximum when A is nega- 
tive, and a minimum when A is positive. 

124. If ^=0, ^ = 0, (7 = 0, then 
F{a J^h,h+k)~ F{a, h) = 

-^ (Lh' + smi'k + SNJik' + Fk') + B,; 

L, M, JSf, F, ^2} denoting the same values as in the preceding 
article. Hence, in order that F{a, h) may be a maximum or 
minimum, L, if, iV, P, must separately vanish, and the sign of 
^2 must be invariable ; and generally, when F{a, h) is a maxi- 
mum or minimum, all the partial derivatives up to those of an 



MAXIMA AND MINIMA. 197 

odd order inclusive must vanish ; while those of the following 
even order are so related, that the sign of the expression in 
which they are the co-efficients of the powers and products of 
h and k remains the same, whatever be the signs of h and k. 
The conditions which will insure this invariability of sign 
when the derivatives are of a higher order than the second, 
are, in the general case, too complicated to be here discussed, 
or even to be of much practical value. 

123. Let it now be required to find the maxima and mini- 
ma values of F{x, y, z), a function of the three independent 
variables x, y, z. 

Keferring to Eq. 7, Art. 121, and in the second member 
stopping with the terms involving the partial derivatives of 
the third order, denoting the aggregate of the remaining 
terms by i?, and carrying F[x, ?/, z) to the first member, we 
have 
F{x + /^ 2/ + k, z-^i)- F{x, y, z) =2^^^ + -^jj ^ + -^^ ^ 

1 /d'F^ , d'F^ , d'F ., ^ d'F ,, ^ dr-F , . 

When A, /.-, i, have tlio very small values which alone are 

admissible in our investigation, the sign of the second moni- 

ber of (1) will depend upon that of the expression 

dF, , dF, , dF . 
dx^'^^dy^'-^dz'^ 

and the sign of F{x -\-h^ y -\- k, :: -\- i) — F[x, y, z) cannot 

remain iiivariabh) for all the possibU^ vahios and oouibiiuitions 

of the signs oi' h, k, i\ uiiU^ss 

(//'' ^ dF. ^dF. _. 
, h - - k-\- J I --=0; 
dx dy dz 



198 DIFFERENTIAL CALCULUS. 

whicli is, therefore^ the first condition necessary to insure a 
maximum or minimum of F(^x,y, z). But, because h, h, i, are 
entirely independent of eacli other, the above condition in- 
volves the three following : — 

^_o ^^-0 ^-0 n 

which Avill determine one or more systems of values for x, y^ z. 

And we now proceed to inquire what further conditions must 

be satisfied when one of these systems, say xz:::^a, y^=^h, %:=ic, 

renders the function a maximum or minimum. 

Substituting these values in (1), and representing the val- 

,. , d'^F d'^F B^F d'F d'^F d'-F „ ,, 
ues whicn ^^, -^, ^^, j-^ -^-^ ^-^, i?, then assume, 

by A, i?, C, A'^ B' ^ C ^ R^, respectively, we have 

F{a + A, & + A;, c + i) — F{a, h, c) = 

-\ (Ah' + Bh' + CP + 2A^hh + 2B^hi + 2C^ki) + B, (3), 
1. z 

the sign of the second member of which, when 7i, k, i, have 
very small values, is the same as that of the expression 

Ah'' + Bk' + OP + 2A^hk + 2B'Jii + 2C'ki (a) ; 

and this sign must be permanent during all the changes in 
the signs and values of 7i, k, i, if F{a, h, c) is a maximum or 
minimum of i^(a::, y, z). 

Expression (a) may be written 

Make s =: - , t = -.; then we have 

P{As'' + Bf" + C+ 2A'st + 2B's + 2C't)] 



MAXIMA AND MINIMA. 199 

and the sign of this last will be the same as that of 
Js' + Bt' + C+ 2 A' St + 2B's 4- 2 C't, 
/ B C A' B' C' \ 

or ^(•'^+3^^+:i+27^' + 27^ + 25V (')• 

The conditions that will make the sign of ih) invariable for 
all possible values of s and t will make that of (a) invariable 
for all the combinations of signs^ and all admissible values of 
^, Iz^ i. To find these conditions, put {b) equal to zero, and 
solve the resulting equation w^th respect to either 5 or t, 
say s. We thus get 

s~—^^^^^±'^^J^{A''—AB)C'-\-2{A'B'~~AC')t-\-B "— ^c|. 

Now, if the quantities A, B, C, A', B' ^ C ^ have such relative 
values, that the quantity under the radical, in this value of 5, 
cannot become positive for any real value of ^, then the paren- 
thetical factor of (/;) will always be positive, and the sign of 
(5) will be the same as that of A. Putting this quantity un- 
der the form 



iA' — AB) h5'' + 2— -^^ ^j5+ , 

^ V A''~AB ^ A'' — AB, 

wo see, that, to make it iiegative for all values of ^, it is neces- 
sary and sufficient to have 

A'' - AB < 0, i.e., AB - A'' > {c\ 

{A'B' - AC J < (7^'" - AC) {A'' - ABA {d\ 

When conditions (c) and (d) are satisfied by ilio vahios oC 
x,i/, Zj deduced from Eqs. 2, F[a,h, c) is a niaxiiuum m- a mini- 
mum; for then the sign of expression (^/>\ and consoipiently 
that of the second member oi' Va\. :>, is jiormanent. and the 
same as that (A\L I'\((, />, c) is (hoi-olore a maxinunn. if ^/ is 
negative; and a mininuini. \[\A is posit ivo. 



200 DIFFERENTIAL CALCULUS. 

Hence the conditions necessary for the existence of a maxi- 
mum or minimum of F^x^ y^ z) are, that the values of x, y, z, 
derived from the equations 

dx ^ dy ^ dz ' 



should make 

d?F /d^F\'2 

> 0, that is, positive (&) ; 



d^F d'F /d^FV 



dx^ dy'^ \dxdy) 
and 

f d'-F d'^F _ d^F d'-F V 
\dxdy dxdz dx^ dydzj 

d^Fy d^d^l (/d'FV d'F^d^l 
dxdz J dx^ dz^- '^ (\dxdy) dx^ dy'^ ) 

A necessary consequence of conditions (c^) and {d') is, that 

/d}F\_d^d'F d'^Fd'F /d'FV 

\dxdzj dx'^ dz^ ' dx^ dz"^ \dxdzj ' 

d'^F d'^F d'^F 
and hence -7-^, 77^? 77~2; ^^st all have the same sign, which 

is negative when F{x, y, z) is a maximum, and positive when 
F[x, y, 2) is a minimum. 

126. If we have a function F{x, y, z . . .) of n independent 
variables, the first condition for the existence of a maximum 
or minimum would be 

dF^ dF^ dF . 

-—7i-\- — A;+ — z + ... = 0: 
ax ay az 

whence, because A, h, i . . ., are independent of each other, 

dF_ dF_^ ^_A 
dx ^ dy ^ dz 

Eqs. 1 determine values x ^ a, y ^r^h^ z ^z c . . .., which may 
or may not produce a maximum or minimum state of the func- 



MAXIMA AND MINIMA. 201 

tion. To decide this question, we should have to examine the 
term 

in the expression for 

F{x-\-li,ij-\-h,z^i..,)— F(x, y,Z'")- 
If, for these values of x, y, z . . ., the sign of this term is per- 
manent, and negative for all admissible values, and all the com- 
binations of the signs of h, k, i .. ., the function is a maximum : 
if the sign is permanent and positive, the function is a mini- 
mum. It would be found, that, to insure either of these states 

r . n • cl'F <^'F d'-F 
of the function, -,-t' -r^^ -tt"***^ must all have the same 
dx^ ay-' dz^ 

sign, negative for a maximum, positive for a minimum. But 

the investigation of all the conditions to be satisfied in this 

general case, in order that the function may be a maximum or 

minimum, is too complicated to find a place in an elementary 

work. 

127. Maxima and minima of a function of several variables 

some of which are dependent on the others. 

Let it be required to find the maxima and minima values of 

the function u = F{x^ y, z. . .) of the ni variables x, y, z . . ., 

which are connected by the n equations 

/i(x,y, z. ..)^0 

/,(^', y, z. ..) = 



(1). 



])y eliminating from a, n of the ni variaMos. by moans of the ii 
given equations, ^6 would become an expHcit finu-tion of }n n 
independent variables, and its niaxinunu or niininuiiu oouKl 

26 



202 DIFFERENTIAL CALCULUS. 

then be found by the method just explained ; but this elimi- 
nation may be avoided, and the determination of the maxima 
and minima of lo greatly simplified, by the process which fol- 
lows : — 

Suppose the variables x, y, z . . ., to receive the respective 
increments h, Ic, i . . . , hj virtue of which the function passes 
from a given state of value to another in the immediate vicinity 
of this : then, if the given state be either a maximum or a 
minimum, we must have 

clF^ cIF^ (IF. 

The partial derivatives of the first members of each of 
Eqs. 1, taken with respect to each of the variables, are separate- 
ly equal to zero. Taking these equations in succession, multi- 
plying each partial derivative by the increment of that varia- 
ble to which the derivative relates, and placing the sum of the 
results equal to zero, we have 

dx dy dz 

dfi n (^fo . df, . 

^'' + i'^ + 5S-*+-=^ (8). 

dx dy dz 

There being a number {n) of Eqs. 3, these with (2) make n^\ 
equations of the first degree in respect to the m quantities 
A, Icji . . .\ hence, by the combination of these equations, we 
may eliminate n of these quantities, and arrive at a final equa- 
tion involving the remaining m — n quantities, and also of the 
first degree with respect to them. To facilitate this elimina- 
tion, let f^i, !^2"'}l^n) denote undetermined quantities; and 



MAXIMA AND MINIMA. 203 

multiply the first of Eqs. 3 by ii^, the second by ^^ • • •; the 
^"^ by iij^] add. the results to (2), and arrange with reference 
to h, h^ i . . .'. we thus get 

dx^ ^'dx^ ^'dx^ '" ^^^dx^^ 






= (4); 



+ • • 

a true equation when F{x, y, z . . .) is susceptible of a maxi- 
mum or minimum, whatever be the values of ^i^ ^^2 . . ., {a.^^. 

Place the co-efficients of n of the quantities h, h, i . . ., in 
Eq. 4, equal to zero : the n equations thus obtained will deter- 
mine ftjj ftg"-; l^n- ^7 substituting these values of fii, ii^-.-j f'„, 
in (4), 01 of the quantities A, Jc,i . . ., will vanish from that equa- 
tion ; and, if the co-efficients of the m — n of these quantities 
remaining in the equation be placed equal to zero, we have, 
including the n given equations, 

f^{x, y, z . . ,) = . . ./„{x, y, z . . .) = 0, 

m equations from which to determine values a, b, c . . ., for the 
w quantities x, y,z..., respectively. Tliis is equivalent to 
equating the co-efiicient of each of the m quantities h. A*. / . . ., 
in (4), to zero ; and these 7n equations, together with the n 
given equations, will make m -\- n equations, hv moans oi' 
which we may eliminate tlie }i indetorminates //j, /r, ....//, . 
and find the m quantities x, y, :: . . . 

It remains to be ascertained whether the sign ot' the expres- 
sion for F[a -\- Ii, h -^l\c-\- i . . .) — F[a, h, c . . .) is invariabK^ : 
and, if so, whether it be [>ositi\(\ which answers to a minimuin: 

« 



204 DIFFERENTIAL CALCULUS. 

or negative, which answers to a maximum. Theoretically, 
this examination is very complicated ; but, for most cases in 
which this method is applicable, the form of the function en- 
ables us to decide at once which of the two states, if either, 
the function admits. 

When m — 71 = 1, or there is only one more variable than 
there are equations connecting them, the case discussed in this 
article reduces to that of an expression which is implicitly a 
function of a single variable. 

128, In the case in which it is required to determine the 
maxima or minima of a function, the several variables of which 
are connected by but one equation, the process may be still 
further simplified. 

Let u = F{Xj y, z . . .) 'he the function, and 
/ix,y, ....) = (1), 
the equation expressing the relation between the variables 
X, y, z . . .: then, by the reasoning employed in the last article, 
we have 

Multiplying (3) by the indeterminate [i, subtracting the result 

from (2), and arranging with reference to h, Jc, i . . ., we have 

dF d/\. , /dF df\ , fdF df\. ^ ^ ,,, 

d^-'£y + {dy-'i}+[-dz^ W- 

Equating to zero the co-efficients of the several quantities 
h,k,i...,we should have, with the given equation, m-\-l equa- 
tions, by means of which we can eliminate {a, and determine the 
m quantities. But from the co-efficients of h, Jc, i . . ., in (4), 
placed equal to zero, we find 



EXAMPLES OF EXPANSION. 



205 



dx 
dx 



dF 
dy 

dy 



dF 
dz 



<v 

dz 



that is, the ratio of the co-efficients of A, in Eqs. 2 and 3, is 
the same as that of the co-efficients of h^ oi i . . . This rela- 
tion will be found to facilitate the determination of maxima 
and minima. 

The examples which follow are arranged in the order of the 
articles in this section under which they fall. 

EXAMPLES. 

1. If F{Xj y) — x'^{a-\- yf, find the expansion of 

{x + hf{a + y-\-hy 
in the ascending powers of h and h. 

'x\a + yf^2x{a + ijfli+Zx\a-{-ijyh 
+{a^yfh+^x{a^y)Vik+^x\a^y)h'- 
+ 3(a + ?/)Vi2A; + (jx{a + y)hk'' + x'-h^ 
+ ^a 4- y)li''h'' + 2xhh' + ]rh\ 

2. If F{x, y, z) = ax"" + by'' + cz'' + 2exy + 2gxz + 2fyz, 
find the expansion of F{x + /^, y + A'; 2 -{- i ). 

'ax- + %-4-C3-+ 2(^7/ + 2(/.i:3 + 2fyz 
4- 2(a.u + gz + r//)A + 2(% +/. -f c.rV.- 

8. Expand 
in the ascending powers of .r and // 



(x + 7if(a + ij + Jcy= 



xr 



"o + «i', +''.^''\^;+ ••• )(''o+?', i + ''^f.j+ ■■• 



206 DIFFERENTIAL CALCULUS. 



— a^h^^a^l^x -{-a^h^y 
+ ^2 (^2^0^' + "^a^h^xy + a^b^y') 

1. Z. o 
+ 

4. Find what values^ if any, of x and y, will render the func- 
tion F{x, y) z=z x'^y -{- xy"^ — axy a maximum or minimum. 

From the equations -^ zr: 0, ^— = 0, we get four systems 



of values, viz. 








X:^{)) 


X =.a\ 


^ = 0) 


a 

" = 3 

1 




\ 


\ 


y = (i) 


2/-0J 


y^a) 


a 



none of the first three of which satisfy the condition 

dx'^ dy' / \dxdy) ^ ^ ' ^' 
and must therefore be rejected. The fourth system reduces 

this inequality to -a^ \ — , wdiich is true, and at the same 
y / y 

time makes both -—^ and -^-.^ positive : hence the values 
ic = -, 2/==Q7 make the function a minimum, and this mini- 

o o 

a' 

mum is — ^• 

5. Determine the values of x and y that will make 
F{x, y) r= e-'-'+y'\ax'' + hy"") 
a maximum or minimum. 



EXAMPLES. — MAXIMA AND MINIMA. 207 

flW rIE' 

-— z= 0, -^- =: 0, 2:ive the three systems of vahies 
ax ay 

y^(j\' y^^iy y = \' 
which we will examine in succession. 

1 he nrst system gives -r— 2 = ^<^; -7-T ^= ^^7 i ^ 7 ' ^^ ^ ' 
hence the values xz=0, y = 0^ make the function a minimum. 
With the second system, we have 

dx^ "^"^ ' dy'^ ' dxdy 

hence the existence of a maximum or minimum depends on the 

relative values of a and h. If 6 is greater than a, ~j-y^ '^rr' 

have the same sign, Avhich is negative, and the function is a max- 

d^^F d'F 
imum; but, if Z> be less than a, ,-.^-; -,-^ , have opposite signs, 

and the second system of values of x and y make the function 
neither a maximum nor a minimum. 
For the third system, we find 

d'F . . d'F ^ , X 1 c?'-^ n 

d^-^-^'^ dy^-"'^-'^'^^ dxdy-''' 
from which we conclude that j:; 1= zh 1, 7/ 1= 0, will make the 
function a maxiinum Avlien a > ?>; but, when a <^h, it has 
neither a maximum nor a minimum. 

6. The equations of two pianos, reforred to roclangular co- 
ordinate axes, are 

/,(.r, y,z)= Ax J^ By +C:: - D = ^\^, 
fix, y, z) = A'x + r/y +C'z-I)' = ^'JV 
It is required to find the shortest distance from the origin of 
co-ordinates to the lino of intersection o( the pianos. 

Let 7'\.r, ?/, z) = x- + //- + ;:- ^;^>^ 

represent the square of the distance from the origin io the 



208 DIFFERENTIAL CALCULUS. 

point of which x, y, z, are the co-ordinates : then, if x, y, z, 
are the same in the three Eqs. 1,2^ 3, the concrete question is 
reduced to the abstract one of finding the values of x, y, z ; 
which^ when subject to the conditions of Eqs. 2 and 3, will 
render F{x^ y, z) 3, minimum. 
By Art. 128, we have 

2x-{-n^A-\-ii^A' — ^ ^ 
2y-\-^,B + f,,B^ = I (4). 
2s + /Xi(7+iU2(7' = ^ 

Multiplying the first of Eqs. 4 by A, the second by B, and the 
third by C, adding the results, then, by (1), we have 

{A'-\-B' +C-'){A,-^{AA' + BB^ + CC^)l^, + 2D = (5). 
In like manner, 

(^/2 _|_ j5^2+(7^')^i + {AA^+BB' + CC^)[^,+ 2D^ = (6). 
From Eqs. 5 and 6, we get the values of ^i, 1^25 ^^^ Eqs. 4, 
when these values of fii, ^^2, are substituted in them, will de- 
termine X, y, z. Multiplying the first of (4) by x, the second 
by y, and the third by z, adding results, and reducing by Eqs. 
1, 2, and 3, we have 

2F{x,y,z)JrI>f^i+D'H = 0; 

from which we get F{x, y, z). In this case, it is unnecessary 
to examine the sign of F{x-\-li, y-^-h, c -{-i) — F{x, y, z), 
when the values of x, y, z, are substituted; for we know from 
the conditions of the geometrical question that the function 
has a minimum. 

7. Required the values of x, y, z, that will render the func- 
tion 

u = x^y^z^ 

a maximum, the variables being subject to the condition 
V z= ax -}- hy -\- cz — k = 0. 



EXAMPLES.— MAXIMA AND MINIMA. 209 



We find 

dx -^ ^ X ^ dy y dz z 

dv __ dv 1. dv 

dx ' dy ^ dz 
therefore (Art. 128) 



ax hy cz k 

p h oh r k 

a p -\- q-\- r'' h p -\- q-\- r^ c p) -\- q -\- / 

These values of x, y^ z, make the function a maximum. For 
we find 

d'^u p du p d'^u q du q d'^^u r du r 

dx^ x^x x'^ ^ dy^ y dy y'- ' dz'- z dz z'- 

and these, because -, = 0, -7- = 0, - .- ^ 0, for the values of 
dx ^ dy dz 

x, yj z, become 

d'^u p d^-u q d'^u r 

dxr'~~ x:'^'~dAf'^~ y' ^' liz' ^ ~ ^ ^^' 
all of which are negative, — a necessary condition for a maxi- 
mum; and, by 2:ettin2: the partial derivatives / , , -^ — ^, ,-^, 
' ' ^ ^ ^ ^ dxdy' dxdz' dydz' 

we see that the other conditions (Art. 12G) to insure this state 
of the function are also satisfied. 

By making a=l,6 = l,c=:l; the above becomes the solu- 
tion of the problem for dividing the number k into three such 
parts, that the product of the jp power of the first, the q power 
of the second, and the r power of the third, t^hall be a maxi- 
mum. 

8. Inscribe in a sphere the greatest parallelopipodon. 

r If a bo the radius of the sphere, the parallolo- 

\ . . '\? 

pipodon is a cube having ~ for its cnlaw 

27 



Ans. 



210 DIFFERENTIAL CALCULUS. 

9. Determine a point within a triangle, from which, if lines 
be drawn to the vertices of the angles, the sum of their 
squares shall be a maximum. 

C The point is the intersection of the lines 

Ans. \ drawn from the vertices of the angles to 

L the centres of the opposite sides. 

A function F{x, y, . . .) of two or more variables may be of 

such form) that it admits of a maximum or minimum for values 

of the variables which make -,-, ^y-, . . . indeterminate or in- 

finite. There are no general rules applicable to such cases ; 
but each one must be specially examined. 

10. What values of ic and y will make 

u = ax'^ -\- {x'^ -\- hi/'^)3 

a minimum ? 

du ^ 2 X du 2 y 

— 2ax + _ ^ 



dx ^3 (aj2 _^ 5^2)r dy 3 (^2 ^ 1^2 f 

For (T = 0, 2/ = 0, these differential co-efficients take the form 
- : but their true values are infinity ; for, if we make y = mx, 

they become 

du ^ , 2 1 du 2 m 

. — ZiCLX —I— — 

dx ^x^{l-\-m''hf dy ^ x^{l -\-m''hf' 

Hence for ^k = 0, and therefore for y = 0, at the same time, 
we have 

du du _ 

dx ^ dy 

For a? = 0, ?/ = 0, we have i^. = ; and no real values of x and 
y can make u negative. Hence t^. is a minimum for x=:0, 
y = 0. 



SECTION XIII. 

CHANGE OP INDEPENDENT VARIABLES IN DIFFERENTIATION. 

129, It is often required in investigations to change dif- 
ferential expressions, obtained under the supposition that cer- 
tain variables were independent, into their equivalents when 
such variables are themselves functions of others. 

Suppose that, having given y^==.f{z\ xz=zq)(z), it is required 
to express the successive derivatives of y, taken as a function 
X, in terms of those of x and y taken with respect to z. 

We have found (Art. 42) 

dy dy dz 

dx dz dx' 
and (Art. 41) 



Hence 













dy 






dz 
dx^ 


1 

- dx ' 

dz 

di/ 


' ''' 


dx 
dy 


dz 

dx' 

dz 






d 


tz 
dx ~" 


d 

'-dz 


dz dz 
dx dx 

dz 


(Art. 


42) 




dhj 


dx 


d^-x dy 








dz'^ 


di- 


dz'- 


' dz dz 










(Mf 


dx 






: 


dhj 
dz' 


dx 
d~z~ 

(dx 


d'j. 

Y 


; dy 

^ ^2 . dz 
' dx 


1 

~ dx' 






\dz 


; 






dz 



211 



212 DIFFERENTIAL CALCULUS. 

So also 

d}y dx dp-x dy 
d^y __ d 'dz^ ~dz dz'^ dz 
dx^ dx /dxV 

d'^y dx d'^x dy 
d dz'^ dz dz^ dz dz 
dz /dxY dx 

\dz 

d^ydx d^xdy\/dx\^ /dxVd'^x /d'^ydx d'^xdy\ 
dz' dz~d? dz)\di) ~ \dz) dz'' [di' dz~'dz''' dz/ dz 

dx\^ dx 

dz) 

d^y dx d'x dy\dx d^x /d'^y dx d'xdy 
dz^ dz dz' dz/dz dz'^ \dz' dz dz' dz 

dx^' 

dz 

d^ 11 d^ 1/ 
In the same manner, we may find -i-^^i T~5 * ' ' Substitut- 

dz^ dz^ dz' 

y—f{z),x = q^{z), 

we have the values of the successive derivatives of y 
with respect to x, in terms of those of x and y with respect 
to z. 

130, Having y =zf(^x), to change the independent variable 

from a; to y in the expressions for -^ - , -^-f . . . 



dx dij d X 
ing in these the values of -^, -~ ^ -^-^ , found from 



CHANGE OF INDEPENDENT VARIABLE. 213 

dy 

<Py^d^)^^±}_dy ,^rt.42), 
dx^ dx dx dy dx dx ^ 
dy dy 

d^x d^x 

__ dy'^ dy dy'^ 

/dxV" dx /dx" 

\dy) \dy^ 

Similarly, 



d^y 

dx'~ 


d'^x d'^x 

d dy^ d dy'^ dy 

dx /dxV ~ dy /dxV dx 

\dy) \dy) 

d'x/dxV /dxV/d^xV 
dy\dy) \dy) W) 


z= 


" (I)' 

d'x dx fd-'xV 
_ dy' dy W) 



Ct 1J d '^ 7/ 

In like manner, wo may find the expressions for ^—4-; i ,V ' " 

These formulas may also be found from those in the preceding 
article, by making 2 = y ; wlience 

dy _-i d-^y _ d'\/f _ dx _ dx d^^x __ d'-x 

dz"^ ' dz' ~ ' dz'~' •' dz~dy' dz- ~dy- '" 
By the introduction of these vahies in the formulas of Art. 

129, they will be found to agree with those just established. 
I5i. Having given y —J\x) (1), 

and also re = r cos. 0, y = /• sin. \^-2), 



214 DIFFERENTIAL CALCULUS. 

it is evident that we may eliminate x and y from these equa- 
tions, and get a direct relation between r and d; and thus r 

becomes a function of d. 

dv d'^ u 
It is required to express the values of -r- ; t-^ • • • ; derived 

dv d'^ T 
from Eq. 1, in terms of — -? -^—r* • • 



also 



By Arts. 41, 42, 


, we 


have 














dy^ 
dx 


__dy dd _ 
~ do dx~ 


dy 

'do 


1 _ 

dx ~~ 


dy 
dd 
dx 




















dd 


dd 














sin. 


dr 
^dd 


+ r COS. d 


from 


Eqs. 












cos, 


dr 

'^do 


— r sin. 


2; 











_ d 
dx 


sin. 6 


do 


-\- r COS. d 


d 

~ dd 


sin. 

COS. 


dr 

'^dd- 


rcos. 
r sin. 


■^c^^ 


c^a;'^ 


COS. 6 


,dr 
do 


— r sin. 6 


dx 

d 



Performing the indicated differentiation, we find for the nu- 
merator of the result 

d" r dr \ / dr 

sin. d — — + 2 COS. d r sin. d V cos. d~ r sin. 

dd^ ' dd )\ dd 

I d'^r ^ . dr \ / . dr \ 

— COS. d --— — 2 sm. ^ r cos. ^ sm. d -—-{-r cos. <9 , 

V dd' dd J\ dd^ J' 

which reduces to r^ 4- 2 { -^ ] — r - — ] 

^ \ddj dd' 

dx dr 
and the denominator, remembering that — ^= cos.d-- rsm.d, 

is 

dr 

COS. d — r sin. 

dd 



CHANGE OF INDEPENDENT VARIABLE. 



215 



Hence 






r2+2 



/drV 
\do) " 



dfr" 



COS. d 



dr 

do 



r sin. 6 



These formulas are used in the appHcations of the differen- 
tial calculus to geometry, where a change of reference is 
made from rectilinear to polar co-ordinates. 

132, Suppose that we have the expressions for — j - ; 

(XjC (Xtj 

found from the equation u = F(x, y) ; but that the variables 

X and y are connected with two other variables, r and d^ by the 

equations x =^ Fi{r, d), y =z F2{r, 6) : then we may conceive 

X and y to be eliminated from these three equations, and u 

to be a function of r and 6, Required the equivalents of 

du du 
the expressions for -^ , -=- , in terms of the derivatives of 
dx dy 

Xj y, and lOj with respect to r and d. 

By Art. 82, we have 

du die dx du dy 

dr dx dr dy dr 

du du, dx du dy 

do ~ dx do "^ dy dO ^ 



(1) 



du du 

7 



and from these two equations the values of ",", "V"? c**^^! ^^^"^ 

dx dy 

found. 

When the e(iuations expressing the relations botwoou 
x, ?/, r, 0, arc r = Fi{x, ?/), == F^{x, y), instead of those 
given above, iluMi 

du du dr du do ] 

dx~' 



du 
dy 



dr (/.<• 
du dr 



-f 



+ 



d') d.v 
du dn 



^2\ 



dr dy d'l dy J 



216 DIFFERENTIAL CALCULUS. 

If the variables x, y^ r, 6, are connected by the unresolved 
equations Fi{x, y, r, 6) = 0, F^(x, y, r, 6) = 0, we proceed 
thus: — 
By Art. 82, 

dl\\ dF.dx dl\dy^ 
dd J dx dd dy dd ' 

fdF\\ dl\ dx dF^ dy ^ ^ ^ 
\dd J dx dd ~^ dy do ' 

in which it must be remembered that ( -— ^ j , ( — -^ J, are par- 
tial derivatives of Fi, F^, with respect to d. 

Differentiating Fi, F2, with respect to r, we get two similar 

dx dt/ 
equations involviiTg ^-, -^ ; and the four equations thus ob- 
tained will determine -^, -^^, -j-, -^ , which must be sub- 
stituted in formulas (1) or (2), Art. 131. 

The following example will illustrate the manner of using 
the above formulas : — 

Given u z=f(^x, y), x =^r cos. d, y =zr sin. d, it is required 

du du . , n r. du du 

to express j^, -^^, m terms of r, 6, ^^ , g^ . 



have 






dx • n ^y a 




dx ^ dy . ^ 
--^ — cos.d, -/ i=sin. ^. 
dr dr 



Hence, by formulas (1), 



dio ^du , . ^ du 

-— = COS. d -^ — h sm. d -,- , 
dr ax ay 

du . ^du , ^du 

-,- = — r sm. d -^ — \- r cos. d -y- 
ad ax dy 



CHANGE OF INDEPENDENT VARIABLE. 



217 



(a). 



dib ^du 1 . ^ du 

whence -^ = cos. d sm. 6 -^ 

ax dr r dd 

du . ^du , 1 ^du 

-^ =: sm. d -= COS. -=- 

dy dr r dd 

To make formulas (2) applicable to this example, we first 
deduce, from the equations a? =: r cos. d,y ^=^r sin. 6, the values 
of r and in terms of x and y. We find 



= V'i«2_|_^2 



tan.-i I ; 



y dd X 

r^ ^ dy T' 



(6). 



-, dr X dr y dd 

whence ^- — -, — - == ^ , — - = 
ttJ:; r dy r dx 

and, by means of these, Eqs. 2 become 

du X du y du^ 

dx r dr r'^ dd 

du y du X du 

dy r dr r'^ dd j 

X 

The relations x := r cos. d, y :=i r sin. d, give cos. ^ = -, 

sin.dz=^^^ by means of which we can pass from formulas (b) 
to (a), or the opposite. 

133, Attention is here called to the necessity of attaching 
their precise signification to the symbols 

dr dr dx dy dd dd dx dy 
dx ' dy dr ' dr ' dx dy dO dd 

which occur in formulas (1 ) and (2\ Art. lot. 

It must be borne iu mind tliat these denote jxirtial dilfer- 

cntial co-efiicients, and (hat those referring to the snmo varia- 

1 1 1 ^>' dx , ^ ^ , , 1 , . • 

Dies, such as , , , , have not to i\uMi i>thor the relation ot 
dx dr 



,"^ to ' , whii'h aro iKm-IvcmI iVoni Ihi^ cqnationy\.r. //") ~ 0. 
(xx ci If 

With reference lo those last, A\e kn^nv tliat one is the reeipi\>- 
28 



218 DIFFERENTIAL CALCULUS. 

cal of the other, or that their product is 1 ; but this is not 

(jv doc 

true for ~- x -^ • The consideration of the meanins; of the 
ax dr 

term ^^ differential co-efl&cient," and the difference between the 
equations connecting the variables in the two cases, will re- 
move all difficulty. In getting formulas {V),x and y were given 
as explicit functions of the independent variables r and ; and 
a change in either r or will produce changes in both x and y. 

d X 

Hence, in the operation of finding -y-^, r, x^ and y vary, while 

d remains constant. In formulas (2), r and 6 were given as 
explicit functions of x and y ; and a change in the value of 
either x ox y will produce changes in both r and d ; and hence 

dv 

the increment attributed to x in getting -y- causes r and d 

also to vary, while y remains constant. That is, in formulas 

dx 

(1), -J- supposes r, x^ and y to vary together, d being constant; 

dr 

while, in formulas (2), -y- supposes ic, r, and d to vary, while y 

remains constant. Thus it appears that 'these two partial de- 
rivatives are obtained on different suppositions in respect to 
the variables which receive increments, and those which 
remain constant. 

In the example just given for formulas (1), we have 

dx dv 

-J- z= COS. (9; and, for formulas (2), ^ := cos. d ; and the product 

dx dr ^ ^ 

-,- -y- = COS.^ d. 

dr dx 

134, Having u = F(x, y, z), and three equations express- 
ing the relations between x, y, z, and three other variables 

r, d, w, it is required to find the values cf -y- , -,- ; -,- , in 
' ' ' ' ^ dx dy dz' 

terms of the different co-efficients of u with respect to r, d, uk 



CHANGE OF INDEPENDENT VARIABLE. 



219 



By Art. 82, 

du _ dio do 
dx ~~ dd dx 
du du do 



du dxfj 
difj dx 



r ^7-,. ^.» I 



du dr 
dr dx 



du d\p du dr 

dy ~~dddy dip dy dr dy 

du du dd du d\p du dr 

dd dz ~^ dip dz ' dr dz 



dz 



(!)• 



The three equations connecting x, y, z, r, 6, -(p, will enable 

, - , . do dd do dr dip ^ ^ ^ . 

us to determine -T- ; -y-; -^ j -^ ;••• ^ ...; and J^^qs. 1; when 
(XiX ay ccz cix ccx 

these values are substituted in them, give us the expressions 

sought. 

By solving Eqs. 1, we can also find the values of , , -^, -^ , 
•^ ^ ^ ' do' dr' dip' 

du du du 



expressed in terms of 
values from the equations 



-^j -T- , -7- ; or we may find these 
ctx cty CLz 



du dio dx du dy 

~ dx dO^ dy dO ~dz dO 



do 
du 



du dz 
dz do 
du dz 



du dx du dy 

dip dx dip dy dip dz da 



> (2). 



du 
dr 



du dx du dy du dz 

— 1 ^ \ _.. 

dx dr dy dr dz dr 



135, Let the relations between the variables x, y, z, 0, if, r, 



X = r sin. cos. i/^, y zi^r sin. sin. if, z = /• cos. i^V). 
From these wo find 



dx d)/ ^, . dz 

y- = r COS. cos. If' . : =z r coi^. sm. w, , =z 
do ' ' (/7 ' ^ ' 



ill 



r sin. 0, 



dj 



■ . . (/// . ^^ dz 

■-r = — r siu. sin. w, ; - = r sm. co^. i/- , = 0. 

dijj ■ ' dip ' (/u- ' 

dx . ^ du . ^ . dz 

- — sill. ^ COS. I/' -y- = sill. ^isin. u- ,=cos.O: 

dr ' dr ' dr 



220 



DIFFERENTIAL CALCULUS. 



and formulas (2), Art. 134, by the substitution of these values, 
become 

du ^ du ^ ^ . du . du 

— z=z rcos.f^cos.o/; — -f-rcos.^ sm.i/j — — rsm.^ — 
dd dx dy dz 



du . ^ . du , . ^ dio 

— -_ — ^ sm. sm. \p — -\- 7" sm. cos. \p — 
d\p dx dy 

du . ^ du , . ^ . du , ^du 

— = sm. d COS. \p — -|- sm. d sm. ip — -f- cos. d — 
dr dx dy dz 



(a). 



J 



From Eqs. a may be found the required values of -y- , -j-, -r-, 

. , p du du du 

m terms oi -,- , -^ , -,- . 
dd dip dr 

Again : squaring Eqs. V, adding results, and taking square 

root, we have r = \/{x'^ + 2/^ + ^^)- Adding the squares of 

first and second of these equations, we find r ^ sin.^ d = x'^-{- y'^ ; 

whence rsin. d = a^(x} -\- y^), sin. d z=:->^(a?^-|-2/^)' ^^^ from 
this, and the last of Eqs. V ^ we find 

tan.^^^l-^ ^^-^^, ^ = tan.-i^^-^ '-^' 

% z 

Dividing the second of Eqs. V by the first, we have 
tan. ip =z ^^ \p=: tan.~ ^ - . 

X X 

Hence we have 

r = V(^' + 2/' + ^'); ^ = tan.-i ^""^ + ^' , ^ = tan.-^ I (20- 

From those by differentiation, we have 

dr X . ^ dr y . ^ . dr z 

^ ■=. - =. sm. d cos. 11;, -- =r ^ = sm. ^ sm. a/;, — z= - = cos.^, 
ax r dy T dz r 

do z 



X 



iC 



cos. cos. -li^ 



c/a; flj2 + ?/2 _j_ ^2 ^^2-_|_ ^- 



J^ 



X 



y cos. d sm. yp 



dy a:' + 2/'-r^' V^-'' + r 



CHANGE OF INDEPENDENT VARIABLE. 



221 



^^ — _ V'(^^+y ') _ _ sm»<^ 
dx x^ -{- y'^ ~~ 



X 



COS. !/> C?li^ 



r sin. d^ dy x^ -\- y'^ r sin. d^ dz 
By the substitution of these values in formulas (1); Art. 134, 
we have 
du __ COS. COS. \p du sin. li) du . _ du 

dx r 



do 



sm. \p du 

; -^ h sm. d COS. 'Kp -y- 

r sm. d dip ^ dr 



du COS. d sin. \p du cos. i/; du 

dy~ r dd^ r sin. d\p~^ 



du 

dz 



sin. ^ du 



du 






(6). 



The values -^, -,^, -— , ffiven by formulas (a), will be found 
dx^ dy^ dz' ^ '' ^ " 

to agree with those given directly by formulas {h). 



EXAMPLES. 



1. Transform 



d'^y (dy\^ dy 

into its equivalent when neither x nor y is independent, but 
both are functions of a third variable z. 

Substitute for -^ '{ and , - their values given in Art. 129, 
and we have 

d'^y dx d^-x dy 
dz^ dz dz^ dz 



dy 



fdxV 
\dz) 



and, multiplying through by 

c?-?/ dx d'x du 

- ' - — '- 4_ 



, dz 
dx^' 



dy 






dz'^ dz 



dz' d 



dz 



jcr)"«>- 



222 DIFFERENTIAL CALCULUS. 

If we make x=^z, this reduces to the given equation. 
Making y = s, (2) becomes 

d'^^x /dx\'^ 

Equation (3) is the equivalent of (1) when the independent 
variable is changed from x to y. 

2. Change 

into its equivalent when both x and y are functions of a third 
variable z. 

( /dyV /dxV )^ /d^y dx d'^x dy\ 

If y z=z z, the above becomes 

( /dxV )-l d'^x 

in which y is the independent variable. 

3. Eliminate x between 

and find what the differential equation is when 6 is the inde- 
pendent variable, and also when y is the independent variable. 
First suppose both x and y to be functions of a third varia- 
ble, z; then the differential equation becomes (Art. 129) 

d'^y dx d^-x dy 1 dy /dx\} /dx\^ 

d? di~'d?dz'^xd'z \dz) "^ ^ \d^) ^ ^ ^^^' 

From x^ = 4:0, we have x = 2(9^, -- := — — — : but-^ = 6~^: 

^ dz do dz do ' 

dx ^_idd d'^x d _xdd ^.xd'^d 1 ^ fdd'^'^ 

• . ft 2 . fj-^ ft 2 ft- 2 

''dz~ dz'dz^~dz dz~ dz'~r \dz 



doc cl'^ oc 
In (2) replacing x, -^, -^^2, ^7 their values, we liave 

1 do d''y _id''d dy _3 fddV dy 3 fddV 

which does not contain x. Making y = 2, (3) becomes 

and if, instead, (9 = 2, we have 

4. Given the relation x = e^, to change the independent 
variable, in the differential expression x^ ^ «? from x to s. 
By Art. 42, we have 

ds\ dx""/ dx\ dx^'/ds \ dx"" dx"-^^ 






.•. —[x'' — -\ — nx'' — ^- z^x''^^ -\ 

ds\ dx'' j dx" dx"-^'^ 

or, writing the first member in an abbreviated form, 

--.n\x'' — ^i=a;" + i -^ (1). 

ds ) dx" dx" + ^ 

Making n = 1, this gives 

d \ di/ d'^y 

l)x /=x'y^ (2). 

ds I dx dx' 

dx 
From X = e% we get . — e' = x ; also wo have 

CCS 



dy __ dy dx dy ^ 

ds ~ dx ds dx 



henco (2) becomes 



„ d'^ II ( d \ dii 



224 DIFFERENTIAL CALCULUS, 

When 71 =: 2 in formula (1), then 

ds I dx^ dx^ 

d}y 
and, putting in this the value of a?^ y-y from (3), 

dx^ \ds J\ds Jds 
The law governing the construction of these equations is ob- 
vious ; and we may write, generally, 



dx"- \ds J \ds J ' ' ' \ds 
The meaning of the operations denoted in the second member 

d d 

of formula (4) is, that if the expressions — — 1, — — 2 ... , 

CIS CCS 

be combined by the rules for multiplication, the result will 
represent, in terms of indicated differentiations on ^, the value 

''f^ ST-- 

5. If we have n =: — -^ ^ , and the relations 

d^ 

dx^ 
x:= r COS. dy y ^ r sin. 0, find the equivalent for p when a 
change of independent variable is made from x to 0, and also 
from X to r. 

When d is the independent variable. 



^~ d'r ^/drV / 
r — „ — 2 ( — —r' 



d'-r /df 

dd''~ \dd 



CHANGE OF INDEPENDENT VARIABLE. 225 



and, when r is independent, 



{•-■(f 



c?^ do dd 

dr'" dr dr 



SECTION XIY. 

ELIMINATION OF CONSTANTS AND ARBITEARY FUNCTIONS BY 
DIFFERENTIATION. 

136. When an equation is given in the form 

F{x,y) = c (1), 

the constant c will disappear on the first differentiation, and 
the successive differential equations derived from (1) will be 
identical with those derived from 

F{x,y) = (2). 

Though an equation may not be given under the form of (1), 
it often happens that one or more of its constants may be made 
to disappear by successive differentiation alone. 

Let {y -hf -^{x — af-r'':=i{i (3), 

and differentiate this equation twice, taking x as the independ- 
ent variable. We find 

(2/-6) J + a;-a = (4), 

<='-')S+(2)'+'=» '='^ 

and thus the two constants a and r of (3) have vanished in 
the two differentiations which lead to (5). A third differen- 
tiation gives 

226 



ELIMINATION OF CONSTANTS AND FUNCTIONS. 227 

From Eqs. 3, 4, 5, and 6, we get 

dy __x — a cf^y __ {x —■ of -\- {b — yf _ r"^ 




dx h — y dx^ ip ~ vY {^ — yf 

dy d'-y 
c^ ^ _ dx dx"- _ Sr^x — a) 
dx-^ y-6 -^ib-yf ^ 

and, by eliminating y — b between (5) and (6), we get 

^|^_3^y/W_Q (7). 
dx'-^ dx\dx^l 

Between (3) and (4), we may eliminate any one of three con- 
stants a, 5; r ; and, by taking these constants in succession, we 
should have for our results three differential equations of the 
first order, each containing two of the constants. By a proper 
combination of (3), (4), and (5), we can arrive at two differen- 
tial equations of the second order, each containing but one of 
the constants of the primitive equation; and between (3), (4), 
(5), and (G), we can eliminate all three of the constants, ob- 
taining for the result a single differential equation of the third 
order. 

It thus appears, that, by differentiation and elimination, Eq. 
3 will give rise, 1st, To three differential equations of the 
first order, each involving two of the constants o, />, r; 2d, To 
two differential equations of the second order, each involving 
but one of these constants; 3d, To one differential equation of 
the third order, from which all of the constants have vanished. 

By means of Eqs. 3, 4, 5, the vahios k^{ a, h, )\ may be ex- 
pressed in terms of .r, //, and the dcM-ivatives oi' y oi' llie tiist 
and second orders. Denoting these ikM-ivatives by //', // ', we 
find 



228 DIFFERENTIAL CALCULUS. 

137* In general, if we have an equation between x and y^ 
and n arbitrary constants, and we differentiate this equation 
m times successively, we shall have, with the primitive equa- 
tion, m-\-l equations, between which we can eliminate m 
constants. This will lead to a differential equation of the m}^ 
order, in which there will be but n — m of the constants ; and, 
as the constants eliminated may be selected at pleasure, it is 
evident that as many equations of the order m may be formed, 
each containing n — m constants, as we can form combinations 
oin things taken m in a set, which is expressed by 

n(n — \) {n — 2) . . . (?i — m + 1) 
1.2.8. ..m 

When the original equation is differentiated n times, we 
should have altogether n-\-l equations, between which the n 
constants can be eliminated ; and, as the resulting equation 
w^ould involve the n^^ differential co-efficient of y taken with 
respect to x^ it is said to be of the n^^ order. The order of 
the highest differential co-efficient entering any of the equa- 
tions at which we arrive, by the steps above indicated, deter- 
mines the order of the differential equation. 

It is worthy of remark, that if any one of the differential 
equations of the m^^ order, obtained by eliminating between 
the first m derived equations, and the primitive equation, m of 
the constants entering the latter, be differentiated oi — m times 
in succession, then this equation of the m^"^ order, and its 
n — m derived equations, would enable us to eliminate the re- 
maining constants ; and the final equation at which we should 
arrive would be the same as that obtained by effecting the 
elimination between the primitive equation and its n succes- 
sive derived equations. 



ELIMINATION OF CONSTANTS AND FUNCTIONS. 229 

To illustrate^ take the equation a=^x — ^- — j^ — , and differ- 
entiate with respect to x. We should find^ after reduction, 

which agrees with Eq. 7. 

The theory of the elimination of constants by differentiation 
is sufficiently simple, and needs but little explanation. It is 
referred to here for the reason that a knowledge of the forma- 
tion of differential equations assists in understanding the more 
difficult and highly important operation of passing back from 
such equations to those from which it may be presumed that 
they have been derived. 

138, Functions known and arbitrary may also be elimi- 
nated by differentiation. 

Let y = a sin. x ; then -^ ^:ia cos. x r= Va- — y^ ' 

an equation which no longer contains the known function 
sin. X. 

Again: suppose 2;=: r/}f-L in which x and y are independ- 
ent, and (p denotes a function of the ratio of those variables 
the form of which is not given, and is therefore called an arhi- 
trary function. 

Make t^^^', then z == ^ {t\ ^^ = .^(1) f = ^ ,r\f). 
y ^ dx ^ dx y 

dy dy y- ^ 

dz , dz ^ 
dx '' dy 



230 DIFFERENTIAL CALCULUS. 

This last equation is true, whatever may be the form of the 

function of- denoted by (jp ; it may be z^^l(-\z— sin. -, or 

s = ey : and for each of these cases the differential equation 
subsists. 

Take the more general case, u=: cp (v), in which u and v are 
known functions of the independent variables x and y, and of 
the dependent variable z, and (:p{v) an arbitrary function of v. 
Differentiating u:= cp{v) first with respect to x and z, and then 

with respect to y and z, and, for brevity, making -p =: p, -j- r= g, 

we shall have 

dto du . , /dv dv 



du du fdv dv\ 

Dividing these equations member by member, we have 
du du dv dv 

dx dz dx~^ -^ dz 



du du d'^ _i_ dv 
dy ^ dz dy ■'- dz 

Clearing of fractions, and making 

p du dv du dv ^ _ du dv du dv r> du dv du dv 

dy dz dz dy^ ' dz dx dx dz ' dx dy dy dx' 

we find that the partial differential co-efficients of the first 
order are connected by the equation 

and this equation is in no wise dependent upon the form of the 
function characterized by qp ; in other words, this function has 
been eliminated. 

139 • Suppose c and c^ to be two known functions of cc, y, z, 



ELIMINATION OF CONSTANTS AND FUNCTIONS. 231 

expressed by c =/(a:, y, z), Ci=/i{x, y, z) • and that, in the 
equation 

F(x,y,z,cf{c),(f,{c,)^ = (1), 

(p, (pi, denote arbitrary functions. Let us see if it be possible 
to pass from (1) to a differential equation which shall not con- 
tain g)(c), q)i{ci), or their derivatives. 
The equations 

^=0 '^-=.0 ^=0 (3) 
dx^^ ' dxdy ' dy'^ 

that we get by differentiating (1), will contain the unknown 
functions q>^ (c), cpi{ci), q)^\c), (jp/^(ci), which, with g)(c), (pi{c^), 
make six quantities to be eliminated between Eq. 1 and the 
five equations of groups (2) and (3), which are generally in- 
sufficient. Passing to the equations 

dx' ~ ' dx'dy ~ ' dxdy^ ~~ ' chf ~ ^^' 

we introduce two additional arbitrary functions cf"'{c)j t/''(^i)> 
and only these two. We shall now have ten equations, viz. 
Eq. l,and those of groups (2), (3), (4), and but eight arbitrary 
functions to eliminate : hence the eliininatioD can be effected, 
and we may have two resulting diirerontial ot|uations of the 
third order. 

We have said, that, in tlie case supposed above, it is gener- 
ally impossible to effect the desired eliminations without pass- 
ing to Eqs. 4. It may happen, howi^ver, that the forms of the 
functions/(.r, ?/, z),J\{^x, y, z), are such that Eqs. 1, 2. 3. will 
prove sufllcient- 



232 DIFFERENTIAL CALCULUS. 

Suppose 2 = (T (x -f- ciy) -^ q}^{x — ay) ; then 

_ = ^'{X + «?/) + 9/(-^ - «y); 

d% 

— - ^i^ a(^' (x -\- ay) — aq}' [x— ay)^ 
ciy 

(J z 

_^ = (f'^ix + ay) + ^/(a:; -a?/), 

^ = a-'cf'^ix + ay) + a-'q/^x - a?/). 
From the last two of these equations, Ave find 

dy'^ dx^' 

140, Suppose that we have two functions, 

F(x,y,z,c,cr(c),cf,{c)..})=0, F,(x,y,z,c,(f:{c), cf,{c)..^ =0, 

in which c is an implicit function of x, y, z, and (y(c), g^i(c) . . .^ 
are arbitrary functions of c. It is proposed by successive dif- 
ferentiations to eliminate c and the arbitrary functions. To 
accomplish this, z and c must be considered as functions of the 
independent variables x^ y ; then, having differentiated the 
given equations a number of times successively with respect 
to x, and also with respect to y, we must eliminate the quan- 
tities 

dc dc d'^c d'^c d'^c ... 

^' d^' dy' d^2' ^^' dp'" ^ ^' 

qp(c), q^{c), (^-(c) . . ., cp,{c), 9^/(c), q,-{c) . . . (2), 
between the given and the differential equations. 
Let m denote the number of arbitrary functions 

t(c), (Pi{c), 92(c)..., 
and n any positive integer ; then, if we stop with partial 



ELIMINA TION OF CONSTANTS AND FUNCTIONS. 233 

derivatives of c and of g)(c), g'i(c), of the n^^ order, the num- 
ber of terms in series (1) will be expressed by 

The number of the arbitrary functions 

will be equal to m{n -{- 1). Again: since each of the given 
equations will give rise to two derived equations of the first 
order, three of the second, four of the third, and so on, the 
number of given and derived equations together will be equal 
to (n -{- 1) (n -\- 2). Hence to be able, in the general case, to 
eliminate c and its arbitrary functions, and their derivatives 
up to the n^^ order, we must have 

(71 + l){n + 2)y ^— ' — ^2 — ' -i-{n-\-l)m,or~^l>m. 

This condition will be satisfied if n = 2m — 1, Avhich will give 
2m( 2m + 1) equations between which to eliminate hn- -\- m 
quantities. The number of equations exceeds the number of 
quantities to be eliminated by m ; hence there will be, in gen- 
eral, m resulting differential equations. 

When the proposed equations contain but one arbitrary func- 
tion, c/i(c), of c, tliey become 

F(.^, y, z, c, (jT (c)) = 0, F,(a', y, z, c, q{c)^ =: 0, 

each of which gives two partial deri\'cMl oi| nations of the first 
order; and Ave shall thus have, including the given oqnations. 
six equations between the quantities 

(h dz (/(■ (/(• 

the ohniin;i(i(Hi oi' the last, live of whi^'h will load to a sinu'lo 
80 



234 DIFFERENTIAL CALCULUS. 

partial differential equation of the first order between the 
variables x, y^ z, of which x and y are independent. 

If there are but two arbitrary functions q>{c), cpi{c) of c, we 
should find that the given equations 

F(x,y,z,c,cp{c),qj^(c)^ =0, F^(x, y, z, c, g)(c), ^i(c)) = 0, 

with their partial derived equations of the first order, making 
in all twelve equations, would involve twelve quantities to be 
eliminated ; viz., 

dc dc d^c d^c d'^c 

^ dx' dy^ dx-^ dxdy^ dy^-^ 

g)(c), (]p^(c), g)^'(c), (jPi(c), ^\{c), (pi{c): 

hence the elimination cannot be effected, except in special 

cases. Passing to the partial derived equations of the third 

order, we should then have in all twenty equations, with 

eighteen quantities to be eliminated; viz., the twelve above 

given, and 

etc a c Oj c a G .. , ^ m / \ 
. ^' d^c^' 'dMf-' dP' ^ ^^^' ^1 *^^^' 

additional: and we may therefore have for our results two par- 
tial differential equations of the third order between x, y, z ; 
the latter being the dependent variable. 

In certain cases, it is unnecessary to make as many differen- 
tiations as have been indicated to enable us to effect the de- 
sired ehminations. Suppose, for example, that the given 
equations contain but three arbitrary functions, 9(c), g'i(c), 
^2(0) : in this case, m = 3, 2m — 1 = 5 ; and, to effect the 
eliminations, it would be generally necessary to form the de- 
rived equations of the fifth order, and we should have for our 
results three partial differential equations of the fifth order 
between x, y, z. But if the arbitrary functions are so related 



ELIMINA TION OF CONSTANTS AND FUNCTIONS. 235 

that g)i(c) =: 9'(c), qp2(c) = cp" {c)j tlie proposed equations be- 
come 

F\x,tj,z,c, (p{c), g)'(c), g)^'(c)j=iO, 

F, I X, y, z, c, q){c), q>'{c), q,'' (c) j == ; 

and these, with their derived equations of the first and second 
orders, make twelve equations, involving the eleven quan- 
tities 

do dc d'c d'^c d''-c 
' dx^ dy^ dx^'' dxdy^ dy^^ 

<r{c),r{c), 9>"(<;), 9"'{o), 9""ic); 

and the elimination will lead to a single partial differential 
equation of the second order. 

If the value c be found, as it may be, theoretically at least, 
from one, say the second, of the equations 
F\x,y,z,c,(p{c),(p,(c)\ =0, F,\x,y,z,c,cp(c),(p,(c)\ z:^ 0, 

and this value be substituted in the first, we should have for 
our result an equation of the form 

F\x,y, z,'ip{x,y, z),^p^(x,y, z)\ = 0, 
which is evidently equivalent to the two proposed equations. 
By Art. 139, we shall generally be unable to eliminate the two 
arbitrary functions i/', i/^i, with this equivalent equation and 
its derived equations of the first and second orders ; but it 
would be necessary to pass to the third derived equations to 
effect the elimination. 

EXAJIFLFS. 
\. Ellniiuato the constant a from the equation 



Vl _ x' + Vl -y'^^a {X ~ y). 



> I — X- dx 



236 DIFFERENTIAL CALCULUS, 

2. Eliminate c from the equation 

x'^-\-y'^ :=: ex. 

Ans. y"^ — 2x2/ -^ — a?^ = 0. 

3. Eliminate the functions e"^ and cos. cc from 

y — e^cos. a; z= 0. 

dx^ ax ^ ^ 

4. Erom y :=. a sin. a:; -\-hco^.x eliminate the functions sin.ic, 
cos. x. 

5. If 2/ = ce^'"-~^^^ prove that 

6. If y = Z^e^^^cos. {nx -f- c), show that 

7. From the equation y = ^ _^ eliminate the exponen- 
tial functions. 

Ans. y2_|_^_ 1^0. 

8. From s = g)(e'^sin. y) eliminate the arbitrary function 

characterized by g). 

dz dz 

Ans. sm. y -7 cos. y ^j- z=.0. 

^ dy ^ dx 

X t] z 

9. From -i, + tij + -^ — lr=0 eliminate the constants 
a, &, c. 

1st Ans. icg -— - -\- x\-— — z-j- =0. 
(Xic^ \dx/ dx 

dH fdz\^ dz ^ 

2d Ans. yz -7—, -\- y [ — ] — z— z=Q. 
^ dy'^^\dyj dy 



ELIMINA TION OF CONSTANTS AND FUNCTIONS. 237 



10. From u ::= xf [ - ] ^ ^{^v) eliminate the functions 



/£> <fi^y)- 






11. Eliminate the functions from 

n ^/{x -\-y)J^xy(p{x — y). 
d^u d^u d^u d^u 2 /d'^u d'^u\ 

dx^ dx^^dy dxdy'^ dy^ x-\-y\dx'^ dy'^J 

12. From 

eliminate the arbitrary functions /, (f, xp. 

, / ^d'^z o<^^A , / <^2 dz\/ dz dz\ ^ 

Ans. x^ ^r-o — V -rr, ] z 4- { z — x- y—- x~ y , =0. 

\ dx^ -^ dy'l ^ \ dx ^ dyj\ dx ^ dyj 

d'^ X d'2J 

13. From the equations -j-.^ =: (f{x, y), ;t f = '^{x, y), elimi- 
nate the variable z ; i.e., change the independent variable from 
z to X. 

Ans. 2(p (x, y) = -y .:, . 

^^ '^^ dx d-y 

dx^ 

14. Eliminate the arbitrary functions from 

dx" ^ dxdy ' ^ dy- ' (/.f ' ^ dy 



DIFFERENTIAL CALCULUS. 



IPj^E^T SEOOIsTX). 



GEOMETRICAL APPLICATIONS. 



SECTION L 



TANGENTS, NOEMALS, SUB-TANGENTS, AND SUB-NORMALS TO PLANE 

CURVES. 

14il. The tangent line to a curve at a given point is 
the limiting position of a secant line passing through that point, 
or it is what the secant line becomes when another of its 
points of intersection with the curve unites with the given 
point. It is now proposed to find the form of the equation of 
tangent lines to plane curves. 

Let y z=zf(^x) be the equa- 
tion of the curve BPQ, and 
take on this curve any point, 
as P, of which the co-ordi- 
nates, referred to the rectan- 
gular co-ordinate axes Ox, Oy, 
are x and y. This point will 
be briefly designated as point 
(x, y). Give to x, taken as the 
independent variable, the in- 
crement Ao?, y wiU receive a corresponding increment a?/, and 

238 




TANGENTS AND NORMALS. 239 

X -\- ^Xj y -\- t^y^ are the co-ordinates of a second point, Q^ 
on the curve; then, \i x^^ ^i, denote the general or running 
co-ordinates of a straight hne passing through P and Q^ the 
equation of this line will be 



or 



^^ ^ iC-j- Air — ^ 

Now, conceive the point Q gradually to approach the point P, 
— ^ will, at the same time, gradually approach its limit -,- = ?/^, 

bt.X CX'X 

and finally become equal to this limit when Q unites with P; 
but then the secant line becomes the tangent line. Hence the 
equation of the tangent line is 

y^~y'^'ix^^^~ ^^' °^ 2/1 - 2/ = y\^i - ^); 

in which -p is the tangent of the angle that the tangent line 

makes with the axis of abscissaa. Calhng this angle t, we 
have 

tan. r = == y' ^ cot. % 
ctx 



COS. r =; dt ■ , - = =b 



1 


1 


dy~ 
dx 


y 


1 





dy 

y' dx 
sin. T 1= =t — y- ' — . = =b -, — 



J' +(:f 



14^. The normal line to a curve at any point is tlio 
straight lino passing tlirough the point at right angles to the 
tangent lino at that point. 



240 DIFFERENTIAL CALCULUS. 

Since the normal and tangent lines at a given point are per- 
pendicular to each other, denoting the angle that the former 
makes with the axis of x by v, we have 

. ^^^ 1_ __1____1__^. 

~ tan. T~ y' ~ dy ^ dy' 

dx 
and, if x^, y^, represent the general co-ordinates of the nor- 
mal line, its equation is 

y^- y z=z - -^^ {x^- x), or y^- y = - -^{x^-x). 

Cot. When the equation of the curve is in the form 
F{x^ y) = 0, or the ordinate y is an implicit function of the 

dF 

Cm 11 cl T* 

abscissa, we have (Art. 84)-^== — hence the equation 

^^ dF 

dy 

of the tangent line becomes 



dF 


-y^dy-"^^ 


i normal 






y^dx = ^' 



143, To find the equation of the tangent line passing 
through a given point out of the curve represented by the 
equation F{x, y) = 0, we should make x^, yi, in the equation 
of the tangent, equal to the co-ordinates of the given point. 
Then, since the point of tangency is common to curve and 
tangent, the co-ordinates of this point must satisfy both the 
equation of the curve and the equation of the tangent : hence 
these two equations will determine x and y, the co-ordinates 
of the point or points of tangency. In the same way, we may 
find the equation of a normal line passing through a point not 
in the curve. 



TANGENTS AND NORMALS. 241 

Now, if we have two curves, of wliicli the equations are 
F{x^ y) = Oj F{Xj ^) — c = 0, respectively, the equations of 
the tangent and of the normal to the first curve will be iden- 
tically the same as those of the corresponding lines to the 
second (Art. 142, cor.). Hence, if for given values of x^^ y^, 
and any assumed value of c, the values of a; and y be deduced 
from the equations 

such values will be the co-ordinates of the points of tangency 
of the tangent line drawn through the point (x^, y^). In like 
manner, the combination of the equations 

F{x,y)-c^{), {x,-x)-^-^-{y,-y)-^ = 0, 

will determine the points of intersection Avith the curves of the 
normal lines drawn from the point (^i, ?/i). 
Since the equation 

is independent of c, it will represent a lino which is the geo- 
metrical locus of the points of tangency of the tangent lines 
drawn from the point (:Ci,?/i), with all the curves which, by 
ascribing different values to c, can be represented by the 
equation F{x,y) — c z= 0. So also 

is the equation of the geometrical locus of the intersections 
of the normal lines drawn through the point {x^,y^) with the 
same curves, irence, if these geometrical loci bo constructed 
from their equations, tluM*r intersection with the curve answer- 
ing to an assigned valued oi\' will be tlie jnunts common to the 
curve and tangents, ov normals, as the case mav be. 

81 



242 



DIFFERENTIAL CALCULUS. 



14:4:. Formulas for the distances called tJie tangent, the 
suh~tangent^ the normal^ and the sub-normal, 

Def. 1. The tangent referred to either axis of co-ordi- 
nates is that portion of the tangent line to a curve which is 
included between the point of tangency and the axis. 

Def. 2. The sub-tangent is that portion of the axis 
which is included between the intersection of the tangent line 
with the axis and the foot of that ordinate to the axis, which 
is drawn from the point of tangency. 

Def. 3. The normal is the part of the normal line in- 
cluded between the point of tangency and the intersection of 
the normal with the axis. 

Def. 4. The sub-normal is the part of the axis in- 
cluded between the normal and the foot of the ordinate of the 
point of tangency. The relation of sub-normal to normal is 
the same as that of sub-tangent to tangent. 

In the figure, let F be the 
point of tangency ; then, with 
reference to the axis of x, P3f 
being the ordinate of F, Ft 
is the tangent, 3It the sub-tan- 
gent, PiV the normal, and MN 
the sub-normal. With refer- 
ence to the axis of y, the 
lines of the same name are 
PT, 31" T, F]Sr% and 3F'N', re- 



\ 


\ 


\ 








M'L 


\ 

"a 


^^ 


-T 

\ 1 , 




4 


^/ 




M 


j?\^' ^ 


/ 


'v 


1' 







spectively. 

,, FM ^ ^^ dy 
Now, ^i-^ = tan. Ftx = --^ , 
' Mt dx 



dx 
or Mt :=^ suh-tangent :=^ y --^ , 

if 



Mt = MF^ = 3IF^, 
dy_ dy' 

dx 



TANGENTS, NORMALS, ^c. 243 

Again: 

^= tan. MP]Sr= tan. Ptx = ^^ ; 
MP ax 

MN:=^ suh-normal = y -~. 
Also Pt'^m+PM = r{^J+r=y^\(^^J+l\ : 



P^ = tangent — y ( 3~ ) "^ •'^ > 



^^^PN^PM^MN = y-^y\-^\^y'\{^^]^\\'. 



\ldyV' 
PN = normal =: y ( -j- J + 1. 

Grouping these formulas, we have 

J~/d£\^ dx 

( — j + 1. Sub-tangent z=y —. 

Normal = y \l~j-\-l. Sub-normal ==?/-—. 



14S. A curve may be given analytically by two equations 
of the form y =^ cp (t), x ^^xp^t), which, by the elimination of t 
between them, may be reduced to one of the form y =zf(^x). 
Without effecting this elimination, the equation of the tangent 
line will bo 

dx dy 

and that of the normal, 

(y, - 2/) 2 + (-,-.'•) J = 0. 

When the co-ordnuito axes arc oblique, making with each 

other an anisic co, the limit of the ratio ' or -/ no louiror 

AX ax 



244 DIFFERENTIAL CALCULUS. 

sin X 

expresses tan. r, but —. — 7- ^. In this case, the investi- 

^ ' sm. (co — t) ' 

gation and the form of the equation of the tangent line remain 

unchanged ; but the equation of the normal line becomes 

1 . ^y 

1 + -^ COS. CO 

y^-y:^-^ ^(^^-^)- 

EXAMPLES. 

1. The equation {x^ — x)x + (?/i — y)y = of the tangent 
line to the circle can be put under the form 

-?)■+('- 1) ==^"- «" 

which, if X and y are variable, and x^ and y-^ constant, is the 

X 1/ 
equation of a circle, the centre of which, having —^ ^, for its 

co-ordinates, is the middle point of the line drawn from the 
point (xy, y{) to the centre of the given circle. The radius of 

this circle is equal to - Vx"^ -j- y'^ . ^"ow, for assigned values 

ofxi, yij the points of contact with the given circumference 
of the tangent lines drawn from the point {x^, y{) must be in 
the circumferences of both of the circles ; and, since (1) is in- 
dependent of r, the circumference of the circle of which it is 
the equation is the geometrical locus of all the points of con- 
tact with the given circumference of the tangent lines drawn 
from the point [x^, y{) to the different circles that we get by 
causing r to vary in the equation x'^ -\-y'^ z= r^. 

2. The general equation of lines of the second order (conic 
sections) is 

u^Ay""-^ Bxy -\-Cx''-{-Dy-\- Ex-\-Fz:^Q: 



TANGENTS, NORMALS, ^c. 245 

and the equation of the tangent is 

{x, — x) {By J^2Cx + E) + {y - y) {Bx + ^Ay + B) = 0, 
which the given equation reduces to 

3. The logarithmic curve is that which has yz=i=^lx 
for its equation. For it we have -^ = ^-; and the equations 

CtX XiCl 

of its tangent and normal lines are 

xla{7j^ -y) — {x^ -x)z=^0, xla{x^ - x) + {y, -y)~0. 

The sub-tangent on the axis of y is expressed by x y- = — 

ux id 

and is therefore constant, and equal to the modulus of the 

system of logarithms. 

4. The logarithmic spiral is a curve having 

I ^ tan. I -j^ , or tan.-i ^ = l^^^ ^ ^ 2 _ jj^^ 

for its equation ; whence 

dy , df/ 

^ij — y ^ + y 1 7 

dx ^ __ ^ ^ dx dy _x^ y , 
x''-]-y'' x'-^-y'"' dx~x~'—y' 

and the equations of the tangent and of the normal are 
{x^ - x) {x + ?/) 4- (//i - //) (// - x) =z 0, 
(•^1 — x) {y — x}— (//i - //) ^x + //) = 0. 
When oji, ?/i, are considered constant, and x, y, are made \o 
vary, these last equations represent two ch'cles, the cirounifor- 
ences of which cut the spiral in the points of contact o{ the 
tangents to tlie 8]iiral whicli are drawn IVom the j>oint ^x^, y^). 

5. Denoting the tangent by 1] sub-tangent by 7;, normal 



246 DIFFERENTIAL CALCULUS. 

by N, and sub-normal by N^, determine these lines for the fol- 
lowing curves : — 

First, The circle : a;- + ^/^ = rl 

Second, The ellipse or hyperbola : -^ zb ^ = ± 1. 

Third, The parabola : y- = 2px. 

r=: 2:r*(aj +1)*, y, = 2x, N=p^ {p + 2xf-, N, =p. 

The sub-normal in the parabola is constant, and equal to the 
semi-parameter ; the sub-tangent is double the abscissa of the 
point of tangency. 

Fourth, The logarithmic curve : x^=: j~ly. 



■=J© + ^''''^«=^^'^=^-^^'"J©^ 



Ixla 



In this curve, the sub-tangent on the axis of x is constant, and 
equal to the modulus of the system of logarithms. 

14:6, The Cycloid is a curve which is generated by a 
point in the circumference of a circle, while the circle is 
rolled on a line tangent to its circumference, and kept con- 
stantly in the same plane. 

Suppose the circle of which C is the centre, and which is 
tangent to the line Ox at the point 0, to roll on this line from 



THE CYCLOID. 247 

towards x. While the point of contact is passing from 
to N, the radius CO, which, at the origin of the motion, was 
y 



O R N B 

perpendicular to Ox^ will turn about the centre of the circle 
through the angle NC'P ; and the generating point will move 
from O to F, describing the arc OP of the cycloid. To find 
the equation of this curve, take Ox, Oy, for the co-ordinate 
axes. Let r = CO = radius of the generating circle ; co = NC'P 
the variable angle ; and x z=z OR, y = PR, the co-ordinates of 
the point P : then 

x:=OR = ON—RNz=z arc PN—PQ =ro) — r sin. w = r (w — sin. u), 
yz=iPR:=C'N—aQz=:r—rcos.cj—r(l — cos.u). 

From y =zr {1 — cos. oa), we have 

^' — y • 1 /-^ 9 1 r—y 

COS. 00 == , sin. w = dr \'lry~y^, co = cos.~^ -; 

and these values of co, sin. w, substituted in the equation 
x:=r {(o — sin. co), give 



r -y 



X = r i COS. -^ j =F \/2ry — y'\ 

which is the equation of the cycloid. The minus sign before 
the radical must be used for points in the arc OPO' whie'li is 
described while the points in the somi-circumrerenc'o OLK 
are brought successively in contact Avith the line O.v : and the 
plus sign must be used for points in the arc ()/>. The point 
O' is called the vertex of the cycloid, or rather the vertex of 
the branch 00' B ; since, by continuing the motion' ot' the gen- 



248 DIFFERENTIAL CALCULUS. 

erating circle on the indefinite line Ox^ we should have an 
unlimited number of curves in all respects equal to OCyB. 
From the equation of the cycloid, we get 



dx \ V dy It — y 



N 2/- — y' ' ' dx 



dy 
Hence the equation of the tangent line at any point is 



_ 12^ — 2/ 



and of the normal, 

2/1 -z/ 



^ ^ (-^1 — ^)- 

N 2r — V ^ ^ ^ 



y 

If, in this last equation, we make i/i := 0, we find 

Xi — x=i Vy{2r — y) =. r sin. w := RN. 

dii d X 

Substituting the values of -^, — , in the general formulas, 

for tangent, sub-tangent, normal, and sub-normal (Art. 144), we 
have for the cycloid 



''=A^j'''-Ai 



y 



y S'lT—'i^ 

N=\/^, N,:=:zs/y{2r-y)] 

which last agrees with what was found above ; and from which 
we conclude, that, if supplementary chords be drawn through 
the extremities of the vertical diameter of the generating cir- 
cle in any of its positions and the corresponding point of the 
cycloid, the lower of these chords will be the normal, and the 
upper the tangent, to the cycloid at that point. 



SECTION II. 

ASYMPTOTES OF PLANE CURVES. — SINGULAR POINTS. — CONCAVITY 
AND CONVEXITY. 

14z7» When a plane curve is such, that, as the point of 
tangency of a tangent line is moved to a greater and greater 
distance from the origin, the tangent line continually ap- 
proaches coincidence with a certain fixed line, but cannot be 
made actually to coincide with it until at least one of the 
co-ordinates of the point of tangency is made infinite, such 
fixed line is said to be an asymptote to the curve. Hence 
we may define the asymptote of a curve to be the limiting- 
position of a tangent line Avhen the point of tangency moves 
to an infinite distance from the origin of co-ordinates. 

To establish rules for finding the asymptotes of curves, re- 
sume the general equation of a tangent line 

!/i-y=^^J^{x,- x) (Art. Ul), 

and find from it the expressions for the distances from the 
origin at a 
These are, 



origin at which the tangent intersects the co-ordinate axes. 



dx 
Xi^= X — y , - = distance on axis o{ x i I), 
-^ dy ^ ■' 

d u 
Vi = ?/ — X ;- = distance on axis oi^ ii rl\. 
dx ^ 

Now, thoro may bo two easels in wliirli asymptotos will ox- 

. , -, , i> ,1 dx. - (/// ..... 

ist: 1st, .l>otli :(•—// , , and // — .r ; , niav roniam hnito tor tlio 
dy ' dx 

32 241) 



250 DIFFERENTIAL CALCULUS. 

values a; = 00 , ?/ = CO . 2d, One of these expressions may re- 
main finite while the other becomes infinite. If the expression 
for the distance on the axis of a; is finite while tha{ for the 
distance on the axis of y is infinite, the asymptote is parallel 
to the axis of y ; and it is parallel to the axis of x when the 
distance on the axis of y is finite, and that on the axis of x is 
infinite. 

Ex. 1. The equation of the parabola is 

2 _ i) dy \/p dx y^2x 

^ "^ -^^^' •'• dJ~ V2S' dy-\/p' 

and, for these values, expressions (1) and (2) for a; = oo, y^=^cc^ 
are both infinite. The parabola, therefore, has no asymptote. 
Ex. 2. The equation of the hyperbola is 

a'^y'^ — &-cc^ ^ — fl^6^, or ^ = =b _ s/x'^- — a^, 

Oj 

dy , hx -, dx x'^ — a^ a^ 

_!. = =t , and X — y --~z^ X — rr: — , 

dx a's/x'^-a dy x x 

d\i 
which reduces to for a; = oo : ?/ — x ~ will also become zero 

^ dx 

when a; z= oo , and ~ becomes i -. Hence the hyperbola has 
dx a 

two asymptotes passing through its centre, and making equal 

angles with the transverse axis on opposite sides. 

Ex. 3. The exponential curve : 

cix 

(XX i i 1 fi 1 

X — y — := X — a^ -^^ X := ± oo lor cc = =t= oo , 

^ dy a'' La la 

y — X ^ ^^ a^ — xa^ la z:zz oo for ic = oo , but =z for a; = — co ; 
^ dx ' 

and, for x =^ — oo , ,^ = a^la z= — =: 0. 

' ' dx a'^ 



ASYMPTOTES OF PLANE CURVES. 251 

Hence the axis of x is an asymptote to the curve, and ap- 
proaches the curve without limit on the side of x negative. 
In this reasoning, we have supposed a> 1. If a < 1, the axis 
of X is still an asymptote ; but, in this case, the curve ap- 
proaches the axis on the side of x positive. 

14:8. An asymptote to a curve may be defined as the line 
which the curve continually approaches, but which it can 
never meet. An investigation, based on this definition, may 
be given that differs somewhat from the preceding. 

Let y z=. ax -\- ^ be the equation of a straight line, and 
y zzz ax -{- ^ -\- V the equation of a curve, v being a function of 
X and y, which vanishes when x and y are made infinite, or, 
at least, when one of these variables is made infinite ; then 
the straight line is an asymptote to the curve. For the formu- 
la for the perpendicular distance from the point {x, y) to the 

• IT . y — ax — B V 

straight Ime is — /—^ = /— ^ — ;, when the point is a 

va^-j-l vcc^ + 1 

point of the curve. Hence when v vanishes, as it does by 
hypothesis, for one or both of the values a; = go , ?/ =: oo , the 
straight line is an asymptote to the curve. 

From the equation y :=ax -\- ^ -\-v, we have "^ = « -|- ^^-~^— ; 

X X 

whence a is the limit of '^ when x and y arc increased without 

X 

limit. Ill general, for these values of x and y, '^ takes the 

X 

dy 
form ^; but its true value is " = ^ . So, also, j^' is the limit 

of ?/ — ar, and a is tlio limit of • ' ; thorefoiv, in general, .-> is 

kXX 

the limit <.^'^ u " .r. 

^ dx 

When the value of « and p tlnis dotonninod are substituted 



252 DIFFERENTIAL CALCULUS. 

in the equation y zzz ax -\- §, it becomes the equation of an 
asymptote to the curve. 

149. When two curves are so related that the difference 
of the ordinates answering to the same abscissa coDverges 
towards zero as the abscissa is increased without Hmit, or the 
difference of the abscissae answering to the same ordinate 
converges towards zero as the ordinate is increased without 
limit, either curve is said to be an asymptote to the other. 

Suppose we have a curve, the equation of which may be 
made to take the form 

y^ax--\-a,x—^^ [-a„_ia^ + a„ + ^ + ^^^ + ^2^+ •■•(!); 

then the curve represented by 

y~ax''-\- a^x""-^ -\ 1- «„_! x^a^ (2) 

will be an asymptote to the first curve. 
So also is that represented by 

2/ = ax" + a^x""-^ -\ \- a^_^x + a^ + - (3), 

X 

and 

y = ax^ ^ a^x^-^ ^ h «n-i^ +«« + - + -^ W- 

X X 

It is obvious, also, that of the curves represented by Eqs. 
1, 2, 3 ... , any one is an asymptote to all the others. 

Example. Find the asymptotes, rectilinear and curvilinear, 
of the curve represented by 

x^ — xy^ + a?/^ = 0, or ?/ =r ± 



\x — a 



The value of y may be put under the form y = ± cc( 1 



X 



and, expanding this by the Binomial Theorem, we have 

a 3a2 5a3 
'Ix %x^ 16^' 



2/ = ±-(i+.l: + '^^ + S^3 + -) (H 



SINGULAR POINTS. 253 

which expresses the true relation between x and y for points 
of the curve far removed from the ori2;in ; for then - is less 

X 
Q 2 

than 1, and the series 1 -j- ^ -f- n-^+ • • ■ converges to a fixed 
finite limit. Whence we conclude that the curve has two recti- 
linear asymptotes represented by the equation y =::^ I x -{--), 

and an unlimited number of curves, having for their equa- 
tions 

which are asymptotes to it and to each other. 

ISO, Singular points of curves are those points which 
offer some peculiarities inherent in the nature of the curve, 
and independent of the position of the co-ordinate axes. 

First, Conjugate or isolated points are those the 
co-ordinates of which satisfy the equation of the curve, but 
which have no contiguous points in the curve. 

Ex. 1. ^2 _|_ 2^2 __ Q Q^^-^ -^Q satisfied only for a; = 0, ?/ = 0, 
and represents therefore but a single point; i.e., the origin of 
co-ordinates. 

Ex. 2. 2f = x\x'^ — a'^). This is satisfied by .r = 0, ?/ = 0, 
and therefore the origin belongs to the curve : but there are 
no points consecutive to it; for values of x between the limits 
x:^ -{- a, x = — a, make y imaginary. Hence the origin is 
an isolated point. 

Ex. 3. a?/' - x' + hx'' = 0. 

Second, Points d^arrSt are those at which the curves 
suddenly stop. 

Ex. 1. ?/ 1= xlx. Hero x — 0, y — 0, satisfy the equation : 



254 



DIFFERENTIAL CALCULUS. 



but negative values of x make y imaginary, 
therefore a point d'arret. 



The origin is 




Ex. 2. y^ie,~~x. If a; be indefi- 
nitely great, and positive or nega- 
tive, y approaches the limit 1 ; but 
if X be indefinitely small, and posi- 
tive, y approaches the limit ; 
while, for negative and very small values of a?, y approaches 
-\- oo . The curve will be composed of two branches, as rep- 
resented in the figure, and will have for the common asymptote 
to these the parallel to the axis of x at the distance 1. 

Third, Points saillant are those at which two branches 
of a curve unite and stop, but do not have a common tangent 
at that point. 

X 



Example. 



From this we find 



y 



!-{-& 



dy^ 
dx 



l + e- 



+ 



ex 




(l+e-)' 
If X be positive, and be dimin- 



ished without limit, both y and 



^y 

dx 



ultimately become zero ; but if x 
be negative, and be numerically 
diminished without limit, we have 

ultimately 2/ = 0, S — \. Hence 



dx 



the origin is a point of the curve at which two branches unite 
having difi'erent tangents ; one branch having the axis of x for 
its tangent, and the other a line inclined to the axis of x at an 
angle of 45°. 



SINGULAR POINTS. 255 

Fourth, Points de rebroussement^ or cusps, are 

points at which two branches of a curve meet a common tan- 
gent, and stop at that point. The cusp is of the first species 
if the two branches lie on opposite sides of the tangent, and 
of the second species if the branches he on the same side of 
the tangent. 

Fifth, Multiple points are points at which two or more 
branches of a curve meet, but do not all stop, or at w^hich at 
least three branches meet and stop. 

Ex. 1. y^ = x^(l — x'^) represents a curve of two branches 
which cross at the origin, at which the equations of the tan- 
gents are y = ' — x, y ^^ x. 

Ex. 2. The equation y'^ := x^ {1 — x'^) is that of a curve 
composed of two branches which meet at the origin, and have 
the axis of x for a common tangent. The origin is a multiple 
point. 

Sixth, A point of inflexion is one at which the curve 
and its tangent at that point cross each other. 

151, We will now establish the analytical conditions by 
which the existence and nature of singular points in a curve, 
if it have any, may be generally recognized ; omitting, for the 
present, the case in which the first differential co-efficient of 
the ordinate of the curve becomes infinite. 

If a curve has either a conjugate point, a point d'arret, a 
point saillant, or a cusp of the first or second species, wo mav 
pass through this point an indefinite number of strai^-ht lines, 
such that, in the vicinity of this point, there is not on one siile 
of any one of these lines for the last three kinds of points just 
named, or on either side for that first named, any point belong- 
ing to the curve under consideration. 



256 



DIFFERENTIAL CALCULUS. 




This is illustrated 
in the adjoining fig- 
ure, in which JijFi g. 1 , 
is a conjugate point ; 
M, Fig. 2, is a point 
d'arret; M, Fig. 3, a 
point saillant ; and Ji", 
Figs. 4 and 5, are 
cusps of the first and 
second species. 
Now, if, for any one of these cases, two points, P, Q, be 
taken on one of these lines, one on each side of the point if, 
and however near to it, these points may be united by a curve 
which has no point in common with the given curve AB. 
Consequently, if u =/(^, y) =: is the equation oi AB, and u 
is continuous, as is supposed, it cannot change sign, except at 
zero : but no values of x, y, belonging to PQ, can reduce u to 
zero ; for, if so, then that point would be common to AB and 
PQ. Hence the values of o^, ?/, belonging to points of PQ, 
make the sign of u constant ; while the values of x, y, belong- 
ing to the point 31, reduce ic to zero. 

Since, then, the value of u at the point 31 is zero, and has 
the same sign at P, on one side of this point, that it has at Q 
on the other, these points being very near 31, u must be a 
maximum or minimum at 31 according as the sign of u at P 
and Q is negative or positive. In either case, we must have 



du du dy 

dx dy ^x 



0. 



Again : denoting the tangent of the angle that the arbitrary 
straight line P3IQ makes with the axis of x by a, the equation 
of this line, which the co-ordinates of 31 must satisfy, will be 



SINGULAR POINTS. 257 

y z=z ax -\-h : whence -~- zzz a ; and, substituting this above, 

we have 

du du 

dx dy 

But this last equation must hold for an indefinite number of 
values for a, since the line P3IQ is arbitrary ; and therefore 
we must have 

du /^ <^^ __ /^ 

dx ' dy~ 

The co-ordinates of the four kinds of singular points under 
consideration should then satisfy, at the same time, the three 
equations 

(. du . du ^ 

Two of these equations will determine values of x and y to 
substitute in the third. If a set of these values x =. x^, y ^=z y^^ 
verifies the three equations, the corresponding point may be 
a singular point, but not necessarily so. 

To ascertain the nature of the point thus determined, let us 

seek the value of -, - , which the equation , -f- ,- ; =: 
ax ax ay dx 

gives under the form . The second differential equation, 

because of the conditions r= 0, ^ == 0, reduces to 

dx dy 

d"U /di/\' d'u df/ d'u 

Suppose, also, that, by the solution oi^ u —/{x, y^ — 0, wo 
have found y =z F(^x) for the equation o( the braiu'h oi' the 

33 



258 DIFFERENTIAL CALCULUS. 

curve on which the point about which we are inquiring is sit- 
uated. The solution of Eq. (a) with respect to -=- gives 

dy_ _ dxdy ^ S \dxdy) ~ do^ Hy^ 
dx~ d^u 

Hence, dy'^ 

I. From the definition of a conjugate point and these equar 
tions, we conclude that the point x^=i Xq^ y z=iy^^^ will be con- 
jugate : fir si ^ if the two ordinate s 

Vy — F{x^ + li), 2/2 = ^(^0 — t^)^ 
are both imaginary ; second^ if the curve at this point has no 
tangent, which requires that 

/(i%Y dhMd'^u 

\d^y) ~"d^^df^^ ' 
unless we have 

d'^u ^ d'^u _ „ d'^u ^ 

dx"^ ~ ' dxdy ~ ' dy^ 

II. The point x=: Xq, y z= y^, will be a point d'arret : first, 
when only one of the ordinates y = F(xq -f- h), y = F(Xq — h), 
is imaginary ; second if the curve at this point has but one 
tangent, which will be the case when the co-ordinates of the 

point satisfy the equation -,-^ = 0. 

III. The point x=: x^^y z^y^^ will be a point saillant : first, 
if to each of the abscissEe xz=Xq-{- h, x z= Xq — h, there is but 
one corresponding ordinate, differing but little from ^/o? or 
if there are two, and but two ordinates, differing but little 
from ?/o, corresponding to one of these abscissse, and none to 
the other abscissa; second, if the curve at the point ic^, y^, 
has two tangents, which requires that we have 

/d^uy dhcd'-u 
\dxdy/ dx^ dy'^ 



SINGULAR POINTS. 259 

IV. The point x^^y^, will be a cusp, when, the first condi- 
tion for a point saillant being fulfilled, the two tangents at 
that point coincide ; which cannot be the case unless 

\dxdy) dx^ dy^ 

152, To investigate the conditions for multiple points, let 

the equation F(^x, y) = in rational form represent the curve ; 

then 

dF dF dy _ ( k ^ qa\ 
dx dy dx 

Since at least two branches of a curve pass through a mul- 
tiple point, two or more tangents may be drawn at that point : 

hence —-, for such a point, must have more than one value. 
dx 

But, since F{x, y) is supposed rational, - -, -— , will each ad- 

CtX OjII 

mit of but one value for the values oi x^^y ^^ which determine 

dii 
the point. Therefore -j- cannot have more than one value, 

OjX 

dW dF 

unless — - = 0, — - = ; and these are the conditions for the 
ax ay 

existence of a multiple point. The equation from which to 

find the values of / is 
dx 

d'F , ^ d'F dy d'F/df/V 

which will give two real values for -^, if, for the values of .r„ 

dx 



dx^ dxdy dx dy 

ivill 
and ?/(), 



d'FV d'Fd-F^ 

3-. -:^T>0; 



^dxdyj dx'^ dy 
and in this case the multiple point is called a double 2^)1 nt. 



260 DIFFERENTIAL CALCULUS. 

— -0 ^'^-0 ^'^-0 
dx'" ' dxdy ^ dy'^ 

then Eq. (h) becomes indeterminate, and we must pass to the 

diflferential equation of the third order, which, after intro- 

ducins; the above conditions, i.e. -— - = 0, — -— r= . . ., is 

dx dx^ 

d'F ^ d'F dy ^ d'F /dyV d'F/dyV ^ , ,, 



dx"^ dx^dy dx dxdy^\dxj dy^\dx 

This cubic equation will s-ive three values for -^, which, if 

dx 

all real, show that three tangents can be drawn to the curve 
at the point (a?o, ^o) • ^^ point is then called a trijple point. 
If Eq. (c?) becomes indeterminate, we proceed to the differen- 
tial equation of the fourth order, and thus get an equation of 

the fourth degree for finding -—] and, in general, if n branches 

of a curve unite in a multiple point, the co-ordinates of such 
point must verify the following equations : 

dF _ ^^_n ^-0 -^-0 ^'^- 
dx ^ dy ' dx'" ' dxdy ' dy"^ ' 

d^-^F _ d^-'F _ d^-^F _ 

dx^^~ ' dx''-Hy~ '"' 'dy^^~~ ' 
and the ?i*^ differential equation of the curve would in general 

determine n real values for — ^. 

dx 

153, If a curve has a point of inflexion, the co-ordinates 

d^y 
of that point must verify the equation -r~ = 0. 

Suppose the equation of the curve has been put under the 
form y = F(x) ; then the difference Ay of the ordinates corre- 
sponding to the abscissae x and x -{-h is (Art. 61) 

Ay = hF' (x) + ^F"(x)+---+ j-^^ i^<«' (x + Oh). 



SINGULAR POINTS. 261 

The difference of the ordinates corresponding to the same 
abscissse of the tangent line at the point [x, y) is Ay^ =: liF'{x) : 
hence, denoting Ay — a?/i by 5, we have 

S = ^ F"{x) + j^ F"-{x) + . . . + ^-^ i^<»'(^ + «^U- 

When A is very small, the first term in the expression for 6' 
exceeds the sum of all the others ; and consequently the sign 
of 5 for points in the vicinity of the point (x^ y) will be con- 
stantly positive, or constantly negative, according as F" {x) is 
positive or negative: hence, if F" {x) does not vanish, the 
curve cannot cross the tangent at the point (x, y\ and there 
can be no point of inflexion. If F" (x) vanishes, then the first 

term in the value of <5 is :p;.-T7 F"'{x\ if F"'{x) does not vanish 

at the same time ; and the sign of this term will change from 
positive to negative, or the reverse, as li changes from positive 
to negative. This can only be the case when the curve crosses 
the tangent at the point {x^y)] and this point is therefore a 
point of inflexion. If F"^ (x) z= 0, then, by the same course of 
reasoning, we prove that the co-ordinates of a point of inflex- 
ion must verify the equation F"" [x) = 0, ttc. Thus, to find 
the co-ordinates of a point of inflexion, wo seek the roots com- 
mon to the equations 

y = F{x), F"{x) = 0, ov/(x, y) = 0, ^^ = 0. 

A system xz=z x^^ y =ry^^ of these roots, Avifl bo the co-ordi- 
nates of such a point, if the first of the dorivativos that does 
not vanish for them is of an odd order. 

lo4. Tliroughout this investigation oC the conditions tor 
singular points, wo have supposed 7*\.r\ and its dorivativos 
for values of .i; and // in the vicinity o( those oorrospoiuling to 



262 DIFFERENTIAL CALCULUS. 

the point {x^, y^), to be continuous. But, if — =00, we may 

CLOG 

readily determine the nature of the point [x^, y^). Under 
this hypothesis, the two quantities F[x^ -\- 7i), F(Xq — 7i), may 
both be real ; or one may be real, and the other imaginary. 

Firstj If both are real, and both greater or both less than 
F{x^), the point [x^, y^) will be a cusp of the first species : if 
one is greater and the other less than F{Xq), the point will be 
a point of inflexion. 

Second, If one of these quantities, say F{x^ — ^), is real, 
and the other imaginary, then, if F(Xf^ — Ji) has but one value, 
the point will be a point d'arret : if F{Xq — h) has two values, 
both of which are greater or both less than F{Xq), the point 
will be a cusp of the second species ; but, if one of these values 
is greater and the other less than F{x^), the point will be 
simply a limit of the curve. 

Tliirdj If each, or but one, of the quantities 
F{x, + h), F{x,-h), 
has more than two values, the point {x^, y^) will be, in gen- 
eral, both a multiple point and a point of inflexion. 

In conclusion, to obtain the co-ordinates of singular points 
of curves, we seek the values of x and y that will reduce the 

differential co-efficients to zero, to infinity, or to -• The na- 
ture of the point is ascertained by inquiring how many 
branches of the curve pass through the point, and determin- 
ing the position of the tangent line or tangent lines corre- 
sponding to the point. 

155, The terms " concave '' and '' convex " are employed 
to express the sense or direction in which, starting from a 
given point, the curve bends with reference to a given line 



CONCAVITY AND CONVEXITY. 



263 



from the tangent at that point. If it bends from the tangent 
towards the line, it is said to be concave, or to have its con- 
cavity turned towards the line ; but, if the sense in which it 
bends from the tangent is from the line, it is said to be convex, 
or to have its convexity turned towards the line. 

To find the conditions of the concavity or convexity of a 
curve towards a given line, take that line for the axis of x, 
and let P, of which the co-ordinates are x and y, be the point 
at which the curve is to be examined with reference to these 
properties. Draw the 
tangent at F: then, 
from our definition, if 
at P the curve be con- 
vex to the axis of x, 
the ordinates of the 
curve for the abscissae 
x -\- h, X — 7i, must be 
greater than the corresponding ordinates of the tangent at P; 
h having any value between some small but finite limit and 
zero. But, if the curve be concave towards the axis of x, the 
reverse must be the case. 

If the equation of the curve is ?/ = F(x), the ordinate cor- 
responding to the abscissa x -\- h is 

7 2 

y + Ay = F{x) + JiF\x)+f:, F-[x) + ... 




+ 



1.2...n 



F^"\x -\- Oh). 



The equation of the tangent to the curve at the point {X, i/) 
is 2/1 - y = F^(x) {x,-x), or y, = A-r) -f .r^ F\x)-xr{x). 
Observing that .r, //, are tlie co-ordinates of the point of tan- 



264 DIFFERENTIAL CALCULUS. 

gencj, the ordinate of tlie tangent corresponding to tlie ab- 
scissa x-\-hh 

Vi + ^y\ — F{x) + xF'{x) + liF'{x) — xF'{x) 
— F{x)^liF'{x): 

hence, if 5 denote the difference y A^ Ly — (y^ -f~ ^^i); ^^ 
have 

« = ^ F"{x) + . . . + ~^^ F^"\x + eh). 

The sign of this difference, when 7i is very small, is the same 

as that of —-F^'i^x), which has the sign of F'' {x) whether 

h be positive or negative : therefore, if F''{x) be positive, the 
curve is convex to the axis oi x; and it is concave if F"[x) be 
negative. 

We have supposed the point of the curve at which its con- 
vexity or concavity was examined to be above the axis of ic, 
or to have a positive ordinate. Had the point been below the 
axis, F" {x) positive would have indicated concavity, and 
F''{x) negative would have indicated convexity. To include 
both cases in one enunciation, we say, ^^ When a curve at any 

d^ It 
point is convex to the axis of x^ y -— is positive at that 

CLX " 

point ; when it is concave to the axis of x, y -— is negative."' 

\XX 

Cor. Comparing this article with Art. 153, we conclude, that, 
when a curve has a point of inflexion, it will be convex to a 
given line on one side of the point of inflexion, and concave 
on the other. 

EXAMPLES. 

Find the asymptotes to the curves represented by the fol- 
lowing equations : — 



EXAMPLES. 265 

1. y^ — x'^{2a — x). Ans. y = — ^ + -c, • 

o 

2. y^ = (cc — of (x — c). Ans. y z= ic — ^- (2a -|- c). 

3. 0^^?/^ =: a^(j?^ — y^). Ans, y =: :^ a. 

4. (?/ — 2x) (y^ — a:;^) — a (?/ — xY -\- 4a^ (^ + ^) ^=- a^- 

Ans. ?/ z= a^, 2/ = - ^ + ^) y = 2^ + 7>. 

o o 

Find and describe the singular points in the curves of which 

the following are the equations : — 

x^ 

5. y = 2 I . 2- There is a point of inflexion at the origin, 

Qi ~]~ X 

and also at the point having a:; = =t a -y/3 for its abscissa. 

6. 2/(<^'' — h^) ^^x{x — ay — xb^. There are two points of 

2a 
inflexion corresponding to the abscissa3 x =i a, x:= — . 

5 

7. y^ z=z(^x — of- (x — c). There is a cusp of the first spe- 
cies at the point of Avhich cc = a is the abscissa. 

8. x^ — ax'^y — axy''- -{■ a^if- =r 0. There is a conjugate 
point at the origin. 

9. mj'^ — x"^ -\- hx'^ = 0. There is a conjugate point at the 

origin, and a point of inflexion at the point having ^ = — for 

o 
its abscissa. 

34 



SECTION III. 



POLAR CO-ORDIXATES. — DIFFERENTIAL CO-EFFICIENTS OF THE ARCS 
AND AREAS OF PLANE CURVES. — OF SOLIDS AND SURFACES OF 
REVOLUTION. 

156* Let the pole coincide with the origin of a system of 
rectangular co-ordinate axes : denote the radius vector by r, 
and the angle, called vectorial angle, that it makes with the 
axis of X taken as the initial line, or polar axis, by 6 ; then 
the formulas by which an equation expressed in terms of rec- 
tangular co-ordinates may be transformed into one expressed 
in terms of polar co-ordinates are x:=r cos. 6, y ^=.r sin. d. 

To express in polar cO-ordinates the tangent of the angle 
that a tangent line to a curve makes with the axis of x, we 

have, calling this angle r, tan. rz=z -—; and hence (Eqs. a, 

ax 



Art. 132) 



sm. 



dr 

dd 



-[- r cos. 



tan. T = 



dr 

cos. d -- 

do 



r sm. 



and from this we may readily find the expression for the tan- 
gent of the angle that the tan- 
gent line at any point makes 
with the radius vector of that 
point. 

Let ilf be the point, P the 
pole, if T" the tangent line, and 
Px the axis of x, from which d 
is estimated ; then 

266 




POLAR CO-ORDINATES. 267 

PMT =: MTx - MPT : 

hence, by the formula for the tangent of the difference of 
two arcs, 

sin. d—- -{- rcos.d 

do ^ . . 
. tan. d 

cos.^- r sin. d 

do do 

tan. F3IT — -. -, —r-~. 

. dr \ dr 

tan. 6 sm. d—- A-r cos. d ] 

1 , \ ^^ / 

' dr . 

COS. d- r sin. d 

do 

This may also be found directly as follows : Take on the curve 

a second point, Q, the co-ordinates of which are r + Ar, d-\- Ad, 

and draAv ifiV perpendicular to FQ ; then MN:=^ r sin. Ad, and 

QJSfz:^ r -\- Ar — r cos. a^.- hence 

,^^,^ rsin. A^ 
tan. NQ3I = . 

7' -{- AT — r COS. Ad 

Now let the point Q move towards 31, The limiting position of 

the secant Q3I is the tangent 3IT, and the limit of the angle 

NQ^Iis the angle F3IT. Call this angle ^ ; then 

rsin. A^ rsin. A^ 

tan. p z=i lim. = hm. 



r 4- Ar — r cos. a^ ^ • ., -^^-^ 

2?'sm.-— -f Ar 



r sin. A^^ 



z^lim ^t 



Jr Sin.- — 

aO ^ aO 

. „A^^ . A^^ 

^ Sin.- ^ sin. ., 

sin. A/9 2 . *^ -^^^ 

The limit of = 1, lim. — == lim. — *" sin. _i 0. 

aO ' A^^ A^y '2 

, ,. Ar . ^ , , (7r , . , do 

and iim. is donotod nv , : thori^loro tan.p=:r , • 
aO ' do ' (//• 



268 



DIFFERENTIAL CALCULUS. 



157* To find the polar equations of the tangent and nor- 
mal lines to a curve; we may assume the equations of these 
lines referred to rectangular axes (Arts. 141, 142), and change 
them into their equivalents in polar co-ordinates ; or we may 
proceed thus : — 

Let r and 6 be the co-ordi- 
nates of the point M; and r', d'^ 
those of a second point, L, in 
the tangent line : then from the 
triangle FLM, making 




FML := r, 



we have 



r _ sin. FLM _ sin. (^ — ^ ^ + ^) 
r'~ sin. PJiZ ~" smTT" 



= sin. (i 



or 



r 1 dr . 

— =r — — sm. 
t' r ad 



>Ocot.r + cos. (^ — 
^ _ ^/) -|- COS. {d — 



(1), 



observing that cot. t =:= 



1 



1 dr 



tan. r r dd 



(Art. 156). Eq. 1 may 



be written 



. d 



z=zr'-^r^m.{d — 0') (2). 



-»r 1 . 1 1 .1 \ dr du , , , 

Making u =. - , u' = ~\ then --— = —-: and hence, by 

^ r r' r^dd dd 

dividing both members of (1) by r, and substituting these val- 
ues, we find t 

du 



u' = u cos. {d — 0^) 



dd 



mn.{d — 6') (3). 



To find the polar equation of the normal at any point of a 
curve, denote by r and d the co-ordinates of If; and by r', 6', 
those of any point, F, in the normal : then 



POLAR CO-ORDINATES. 269 

PR &in. P3IB . /7t 

sm. ( - — T 

\2 

T 

therefore — = sin. (6' — 0) tan. t + cos. (0^ — 0) 

rdd 
= sin. («'-«)^ + COS. («'-«) (4), 

which may be written 

d ^ dr 

r' -— r COS. (d — 6') =: r — . 

do ^ ^ dd 

Adopting a notation like that in the case of the tangent^ (4) 
becomes 

dft 

u' = u COS. id' — d)—u'^-~ sin. (d' — 6) (5). 

du / V / 

1S8. Let P be the polar point to which is referred the 
curve EMS, and through P draw -~_^ \ 
NT perpendicular to the radius rvv"'7^ 

vector PM; then 3IT being the T '"^^ 

tangent, and UN the normal, to y\/^.^^S^ 

the curve at the point J/, the hues "^^.j/x ] \^^ 
MT, PT, 3IN, and PN, are, re- ^^""-^ \/\ "" 

spectively, the polar tangent, sub- ^^--^^X 

tangent, normal, and sub-normal. ^^t 

To find tlio formulas for the lengths of these lines, put angle 

Pil/J' = /), and resume the equation tan.p'= r (Art. loGV 

dr 

, . do 1 ^ r 

makmg - = -- : whence tan. fi — —, from which wo find 
dr r' r' 

COS. t> = - 7 , sm. p := -'- , 

Then the triaiii;lo /"JA'/'uives 



270 DIFFERENTIAL CALCULUS. 

FT = T, = PMtd^u. F3IT:= rL = ^r'^, 

r' dr 



cos. PMN sin. PMT ^ "^ ^ "^ ^J/ ^ 



Pi\^ =: iV; = Pi!f tan. P3IN^ r cot. 5 == r - = r ' = ± ^^. 

^ r dd 

The polar sub-tangent is considered positive when it is on 

the right, and negative when on the left, of the line PM; the 

eye being supposed at P, and looking from P toward 31. The 

dv 
sign of the sub-tangent will then be the same as that of -r-; 

that is, positive when r is an increasing function of d^ and neg- 
ative when r is a decreasing function of d. 

159, An asymptote to a curve referred to polar co-ordi- 
nates is a tangent line, the polar sub-tangent to which remains 
finite when the radius vector of the point of tangency becomes 
infinite. Hence, to find the asymptotes to a polar curve, we 

d H 

must seek the values of d, which make r infinite while r^ ^- 

' dr 

remains finite. If m be a value of d which satisfies these con- 
ditions, the asymptote may be constructed by drawing through 
the pole a line, making, with the initial line, the angle m, and 
another line at right angles to this through the same point ; 
laying off on the latter, to the right or to the left according as 

r^ -r- is positive or negative, the distance represented by r^ -^, 

and through the extremity of this distance drawing a line par- 
allel to the first line. The line last drawn will be the asymp- 
tote. 



POLAR CO-ORDINATES. 



271 



Example. In the hyperbolic spiral, so called because of the 
similarity of its equation r =: -, or rd z= a, to that of the hy- 
perbola referred to its asymptotes, we have 






a. 




do ^2 do 

— = — — : . * . r — 
dr a dr 0'^ a 

Hence the sub-tangent is constant, and equal to — a : but 

0z=O gives r ^oo ; whence the line parallel to the polar axis at 

the distance from it equal to — a is an asymptote to the curve. 

This curve, beginning at 
an infinite distance, contin- 
ually approaches the pole, 
making an indefinite num- 
ber of turns around without 
ever reaching it. 

160* When the curve in the vicinity of a tangent line at 
any point, and the pole, lie on the same side of the tangent, 
the curve at that point is concave to the pole ; but, if the 
curve and the pole lie on opposite sides of the tangent, the 
curve in the vicinity of the point of tangency is convex to 
the pole. 

Let Pp IZIJ9 be a perpendicular 
from the pole on the tangent to the 
curve at the point ((9, r) : then it 
is plain, that, if the curve at this 
point is concave to the pole, j^) will 
increase or decrease as r increases 
or decreases ; that is, ^) is an in- 
creasing function of r: and, on 
the contrary, if the curve is convex to the polo, j) is a dooroas- 
ing function of r. ITencc, when the curve is conoavo to the 




272 



DIFFERENTIAL CALCULUS. 



,<&>,.. , , dp . 

pole, -T- must be positive ; and, when convex, --- must be neg- 
dr dr 

ative (Art. 51). 

Cor. At a point of inflexion, the curve with reference to the 
pole must change from concave to convex, or the reverse. 



Hence, for a point of inflexion, — 

dr 

We have (Art. 158) 

sin.PJ/r:=sin.5i= 



or 



Vr2 + r 



■• J 



'■+||T 



p =: r sin./3 = 



J- 



+ 



-.r 1 1 . dU 

Make z^ = - ; then -z- = 

r' do 



1 



dry F 
do 
J_ dr 



1 + i- 

r' ^ r' 



dr\' __ Jdu\\ 
d')) ~ "^ \do) ' 



1^ 



— -,/2 



= u' + 



1 dp / d- u\ du 

and — -V ^7 = h^ H — r-> ~^r. j 



hence 



'du 

Jo)' 

dpdd^^dp^_^^,/^dhi 

do du du \ 

dp dp du _ \ djD p 

dr~dudr~~r^ du~ r^ \ "^ dO'^ 



do'' 



d^^u 



Therefore, at a point of inflexion, u -f- -— will generally change 
its sign. 

161, Differential co-efficient of 
s the arc of a plane curve. 

Let F{x, y) = ov y =/{x) 
be the equation of the curve 
HFS referred to the rectangular 
axes Ox, Oy ; and take, in this 
curve, any point, P, of which the 
co-ordinates are x, y. Denote the length of the curve estimated 







/ 




Q 


s 




'IS 


^ 


P' 


^ 


p 




/ 




^ 


L jy 


I i« 


L' ^ 



DIFFERENTIAL CO-EFFICIENTS OF ARCS, 273 

from a fixed point to the point P hj s ; then, if x be increased 
by MM' =r Aic, 5 is increased by the arc PP' = as, and it is 

As 

required to find the limit of the ratio — , or the differential 

A 00 

co-efiicient of the arc s, regarded as a function of x. 

The tangent line to the curve at the point P meets the or- 
dinate P'M', produced, if necessary, at Q ; and makes, with the 
axis of Xj an angle of which the tangent, sine, and cosine are 
respectively 



Now, if, within the interval ax, the curve is continually con- 
cave or continually convex to the chord PP', it is evident 
that arc PP' > chord PP', and arc PP' <PQ+ QP'. 

,_ . PN 

But chord PP'^s^Ax' + Ay\ PQ = TT^Tt^^^V^I+V^'; 

COS. QPN ' ^ ' 

QN—PNi^ii.QPN=y'Ax; .-. QP' — y'Ax—Ay: 

hence, substituting in the preceding inequalities, we have 

Asy y/Ax'^ -\- Ay\ As<^AxVY^^\^y^- -\-y'Ax— Ay. 

Therefore 



r > I 

AX \^ 



. , A?/^ AS ^ A?/ 

1 + '-.y : — < Vi -4- y'' + y — - - 

' AX^ AX ^^ ^ ^ y ' ^ AX 



At the limit, the second member of each of these inequalities 
reduces to Vl -{-y''^ = Jl + T-- j : hence we have 



A s __ (is , fdi/y 

for the differential co-efficient of the arc regarded as a func- 
tion of the abscissa. This must be understood as oxprot^sing 

ds 
only the absolute value of -, for the arc may bo an inoroas- 

35 



274 DIFFERENTIAL CALCULUS. 

ing or a decreasing function of the abscissa, according as it is 
estimated in the direction of x positive or x negative : hence 
the general value should be written 



§=-'^•=-^■+(1/. 



the sign -|- to be taken if 5 increases with cc, and the sign 
in the opposite case. 
Cor. 1. Since 

PP' ■ s/Ax' + ty'' 

= lim ^"^ = 1 



J 



1 + 



9 ' 



and the arc PP' is always included between the chord PP' and 

A5 

the broken line PQ-\- QP', it follows that lim. — — = 1 ; 

\ ^x"- -[- ^y'^ 

and hence, when the arc is infinitely small, it and its chord 
become equal. 

Cor. 2. Squaring both members of the equation 

dx ( \dx) 

and multiplying through by ( — ) , we find 

- \ds) -^ 

Now, if X and y are both functions of a third variable s, then 

dx 

dx dx ds ^ ^ dx dz 

dz ds dz' ' * ds ds 

dz 



DIFFERENTIAL CO-EFFICIENTS OF ARCS. 275 

dy _dy ds ^ dy dz 

dz ds dz' ' ' ds ds ' 
dz 

doo dii 

These values of -, , ~-^ substituted in the preceding equa- 

kAjS CtS 

tion, give 

ds _ Udx\^ (dyVlh 
dz ~ l\dz)^\dz) \ ' 

162, Denote by a and § the angles that the tangent line at 

the point P (figure of last article) makes with the axes of x 

and y positive ; then 

AX 1 dx 

COS. a — hm. , = , ^ — — (Arts. 41, 161), 

VAx'' + Ay' Vl-\-y^' ds ^ 

Av dy dx dy 

COS. ^ = hm. ,- -^ ~-^^^ — ^ (Art. 42) ; 

VAx^ + A?/^ dx ds ds 

or, more generally, writing the sign d= before \^ ax- -\-Ay'\ 

dx ^ , dy 

COS. a := =±= -,- , COS. p = =h — - 

ds ds 

in which the upper sign or the lower sign is to be used ac- 
cording as the angle is that made with the axis by the tangent 
produced from the point P in the direction in which the arc 
increases from that point, or the opposite. Tliis refers to the 
algebraic signs of cos. (X, cos. /5: their essential signs are deter- 
mined by combining their algebraic signs with the essential 

„ dx dy 
siffns 01 - , ■ . 
^ ds' ds 

163. Dillerential co-eHicicnt of an arc referred to polar 
co-ordinates. 

For the transformation o'i rectangular into polar co-ordinatos, 



276 



DIFFERENTIAL CALCULUS. 



we have x z=zr cos. d^ y ^^r sin. d. We also have (Arts. 42, 
161) 

ds ds dx dx I (dyV' __ I /(ia^\ '■^ fdy^ 



do dx do 



dx 



dd 



^ dx dr . dy . dr 

But -- =: COS. d- r sm. (9, — - == sm. d—--\-r cos. 6; 

dd do ' dd dO^ ' 



therefore 



ds 

dd 



=J 



r^ + ( -— ) : and, in like manner, 



_ds dd _ \ 
~'dddr~^ 



dd 



ds 

dr dd dr N ' \c?ry 
Cor. When § is the angle included between the radius vec- 
tor of a curve at the point (r, d) and the tangent line at that 

dd 
point, we have (Art. 156) tan. ^ =: r — ; and hence 

(xr 



dd 
dr 



and 



sm. p = 



cos. p = 



J'+Kf" 



N ^ W dr 



dr 

ds 
dr 

1 dr^ 
ds' 



ds 



164, Differential co-efficient of the area of a plane curve. 

The area enclosed by the arc 
^ ^y^ BP of a plane curve, a given 

[P' ^ ordinate AR, the ordinate of any 
point P of the curve, and the axis 
Ox^ is obviously a function of the 
abscissa of P, since the area va- 
ries with the position of this point 
on the curve. 
Let ARPM = u, and give to vC,the abscissa of P, the increment 




^"o 



DIFFERENTIAL CO-EFFICIENTS OF AREAS. 277 

MM' = ^x ; then MPP'M' = ^u is the corresponding incre- 
ment oiu. and we are to find the limit of the ratio — ■. 

LX 

Through P and P' draw parallels to Ox, and limited by the 
ordinates PM, P'M' ; and suppose a a; to be so small, that, be- 
tween these ordinates, the ordinate of the curve constantly 
increases or constantly decreases. The rectilinear areas 
MPP'M', 3IN'PM', have y^x, {y + Ay) ^x, for their respective 
measures, and the curvilinear area MPP'3P is constantly in- 
cluded between these two ; that is, 

Mi^yLx, Lu <i^{y -\- t^y) Lx : 

whence y <^ — <^y -\^ i^y ; or, by passing to the limit, 

A X 

-—- z= y du = ydx. 
ax 

If the co-ordinate axes are oblique, making with each other 
the angle oj, the above demonstration still applies, observing 
that then the area mo lies between the areas of two parallelo- 
grams, the sides of which are parallel to the axes ; and, since 
the area of a parallelogram is measured by the product of its 
adjacent sides multiplied by the sine of the included angle, wo 

should have --- — y sin. co. 
dx ^ 

165, When the curve is re- — -->v.\. 

ferred to polar co-ordinates, the \^V\ /^ 

area considered is a sector cm- I yy\. 

braced bv a o-iven radius vec- /I //^\\V 

tor PR, the radius vector P^I o'i ^^^i>^ \ x 

any point (<9, r), and llie arc B3I. ^^\ \ 

Denote this area, which is a func- ^"""\\. 

tion of d, by u ; let be increased '^ 

by AC), by which the point JYniovos io aS', and u is incroasod by 



278 



DIFFERENTIAL CALCULUS. 



the sector P3//9z= A zi; and suppose ^^ so small^ that the radius 
vector of the arc from Mio S is constantly increasing or con- 
stantly decreasing. With P as a centre, and the radii vectores 
F3I^= r, FS^=^ r' ^ as radii, describe the arcs MIy SK, limited 
by these radii vectores. We have 

sector PJ/7 < a?^ < sector P>SX; 



or, since sector PJ/J =7^ r-A^, and sector P/S'^ 



r'^^^, 



--r-A^< ^u <-r^2^6'; 



1 



< — < T' 

^ A^ ^ 2 



But, at the limit, r' becomes equal to r: hence 



du 1 , 



du 1= -r'^dd. 



y 



The equations ic=rrcos./9, ?/ = rsin. (9, give - = tan. (9; whence, 
by differentiating with respect to 6^ 



dy dx 

'^dd~ydo 



cos.'-^(? 



r^ COS."! 
cos.'-(? 



= 7-, 



^(^xdy — ijdx) zzz-r'^dd; 

an expression in terms of rectangular co-ordinates for the dif- 
ferential of a polar area that is of frequent use. 

IGG, Differential co-efficient of the volume of a solid of 
revolution. 

If V represent the volume 
generated by the revolution of 
the plane area ABPM about the 
axis Ox, and x be increased by 
3I3I' z=i ^x, the corresponding in- 
crement AFof Fwill be the vol- 
ume generated in the revolution 
by the area MFF'W. Now, if 
A .r be so small, that y increases constantly from P to P', A V will 




DIFFERENTIAL CO-EFFICIENTS OF VOLUMES. 279 

be included between the volumes of the cylinders generated 

by the rectangles 

MPNM, MN'F'M'. 

Hence, denoting MP by y, and M'P' by 7/1, we have 

Tiy^ ^x < aF< Tty'^ acc, or ny'^ < - — <^ Tty"^] 

aV 
that is^ — is comprised between the two quantities Tty'^, tt?/^, 

the second of which converges to equality with the first as 

AX diminishes. Hence, at the limit, 

dV 

-^-zzzTty"^, dV^^ny'^dx. 

(XX 

107, Differential co-efficient of the surface of a solid of 
revolution. 

Let s represent the arc PP (figure of last article), and S 
the surface generated by the revolution of this arc about the 
axis Ox. For the increment M3P ^=Axofx,s will be increased 
by the arc PP^ = A5; and S, by aS :=^ the surface generated 
by AS. When AX is sufficiently small, the surface A^S^will be 
comprised between the surface of the conical frustum gener- 
ated by the chord PP^, and the surface generated by the broken 
line PQP'. The surfaces generated by the chord and by the 
broken line are measured by 



/ du \ ^ d}/ 'dii 

— "IrttjAy — ny^Ay^"^: 
hence, ostabHshing tlu^ iiunjualilios. and dividing through by 
AX, wo have 



280 DIFFERENTIAL CALCULUS. 



:-!<-(-*+i"U'^+-(i)'"-(2)'- 

dy Ml 

At the limit, the second member of each of these inequalities 



becomes equal to ^ny A'^ -\- (y^ ) • hence 



,. ^S dS ^ I /dy\^ ^ ds 

dS = 27iyds. 



SECTION IV. 



DIFFERENT ORDERS OP CONTACT OF PLANE CURVES. OSCULA- 

TORY CURVES. — OSCULATORY CIRCLE. RADIUS OF CURVA- 
TURE. 



168. Suppose y =: i^(x), y =/(x)j to be the equatioiis of 
the two curves RFJSf, R'PN'^ 
which have a common point P; 
and let ns compare the ordi- 
nates M'N, II'N', of these 
curves corresponding to the 
same abscissa OM' =: x ~\- h, 
differing but Httle from the 
abscissa 031 =:x of the point P. 
We have 

3rJSr= F(x + h), M'N' =f{x + 70 : 
NJSf' =z F(x + h) —f{x 4- A). 

Developing each term in the vahio oi NN' by the formula oi 
Art. Gl, observing that F{x) =/(x*) by hypothesis, we find 




dx dx 



1. 



■Li 



dx- 



d"F d'\f^ 



"^ 1.2.7.7i\(/.f" dx") "^ 1.1^...;/-}- 1 \dx" + ' dx" + '/ 



the hist term of which may he writton. — 

30 2S1 



282 DIFFERENTIAL CALCULUS, 






1.2...71 + 1 n-^2 



1.2...7^ + l 
H being a quantity that vanishes witli h: hence 

\dx dxj^ 1.2\dx' dxy^ 

j^n + l / gn + lj^ ^n + y 

'^1.2.,.ni-l\dx^ + '~ dx- + ''^ 
If, in addition to F(x) = fix), we have —-=—-, the curves 

have a common tangent, PT, at the point P, and are said to 
have a contact of the J77^st order: and if, at the same time, 

d^F d^f 

-T—^ ■=■ y^T,, the contact is of the second order; and, generally, 

the contact is of the n^^ order if n denotes the highest order 
of the differential co-efficients of the ordinates of the two 
curves that become equal when in them the co-ordinates of the 
common point are substituted. 

169, When two curves have a contact of the n^^ order, no 
third curve can pass between them in the vicinity of their 
common point, unless it have, with each of the two curves, a 
contact of an order at least equal to the ?^^^ For y =z F[x), 
y z=f[x), being the equations of two curves, BPN, R'PN' 
(figure of the last article), which have at the j^oint P a contact 
of the n^^ order, let y = q^(x) be the equation of a third curve, 
Pi"PN" ^ passing through P, and having with the first curve a 
contact of the ?7i*^ order, m being less than n. Then, by the 
preceding article, we should have 

dF_dq) d'^F __d^ 



CONTACT, OSCULATION, ^'C. 283 

li and i?i being quantities which vanish with h: hence 

dr^+^F d^+y 

JSW h^ - '" dx^ + ' ~ d^+' "^ ^ 

X 



JViV^// (m + 2) . . . (7i + 1) d"'-^'F_d'- + 'cp 

~d^^^^^' ~ ~±^^^' "^ ^' 
Since, as h converges towards the hmit 0, B and 7?^ converge 
towards the same Hmit, and reach it at the same time 7i does, 

and since n > m, it follows that the ratio ^r^r=r can be made 

as small as we please by giving to h a sufficiently small value ; 
that is, when h is a very small quantity, JSfN^ will be less than 
NN^^, and the curve y = q){x) cannot, in the vicinity of the 
common point F, pass between the curves 7/ :=iF(x), y ^:zf(x). 

It is evident that this reasoning holds when li is negative as 
well as when it is positive. 

Cor. When li is sufficiently small, the sign of the expression 
for iVW (Eq. I) will be the same as that of A" + \ and will 
therefore change with that of 7i if n be oven, but remain in- 
variable if n be odd. Hence, if two curves have a contact of 
an even order, they will cross each other at the point of con- 
tact, but not otherwise. 

J 70. Osculatory curves. If llio form of the function 
Fix), and the constants which enter it, are given, the eipiation 
y = F(.r) represents a curve fully determined in respect io 
species, magnitude, and position; but if the form ot" the func- 
tion only is known, the constants which outer it being arbitrary, 
the species of curve is all that the equation determines. Tliiia 
the ecpiation y — h-Az s^ r'- — (.r — aV, when r, (T, /).are lixed in 



284 DIFFERENTIAL CALCULUS. 

value, represents a circle that is completely known ; but if r, a, 
and h are undetermined^ the equation may represent every 
possible circle lying in the plane of the co-ordinate axes. 
It is then the equation of the species "■ circle." 

When a curve of a given species has a higher order of con- 
tact than any other curve of that species with a given curve^ 
the former is said to be an osculatrix to the latter. 

Suppose /( 2^1, ?/i, a, &, c. . . ) = (1); involving n-\-l arbi- 
trary constants, to be the equation of the species of curve that 
is to be made an osculatrix to the curve of which y = F{x) (2) 
is the equation. By means of the 7i -\- 1 constants in (1), we 
can satisfy the n -}- 1 equations 

^ dy ^dy^ (Py^ _(Py^ d^y _d^y, 
^ '^^' dx dx' dx'' dx''""' dx^"' dx- ^ ^' 
or, in other words, these equations will determine the values 
of a, &, c . . ., which, substituted in (1), will make it the equa- 
tion of a curve having a contact of the n^^ order with the 
curve represented by (2) ; and it will be an osculatrix, since, 
in general, a higher order of contact cannot be imposed. We 
conclude from the above, that the number denoting the order 
of contact of an osculatory curve is one less than the number 
of constants entering the equation of the curve. 

Example. The form of the equation of a straight line is 
yz^ax-\-h; and since this equation contains two constants, 
a and 6, w^e may so determine them as to cause the line to 
have a contact of the first order with a given curve at a given 
point. Suppose y = F{x) to be the equation of the curve, 
and that ic = m, y = 7i, are the co-ordinates of the point ; then 
the equations to be satisfied are 

am + 5 = F{m), a = F^ (m), 
which determine a and b. 



OSCULATORY CIRCLE. 285 

17 !• Osculatory circle , or circle of curvature. 

Assume the co-ordinate axes to be rectangular, and let 
y := F{x) (1) be the equation of the given curve; then, since 
{x^ — «)^ + (?/i — hy ^p^ (2) is the general equation of the 
circle, and contains three constants, the osculatory circle will 
have, with the given curve, a contact of the second order. 
From (2) we get, by two successive differentiations, 






(3); 



a^-a + (y-h)± = (i,l+[±)+{y-h)^^=0 (5): 



and, because the circle is to be the circle of curvature, we 
must have 

ctjc ajc J ccjo ctoc 

(Jii (J 11 
These values oiy^^ -, -, -V^, substituted in Eqs. 3, give 

dx ' ^ \dxl ^ ^-^ ' dx 
therefore 

^ d^j dry ^ ' 

dx'^ dx'^ 

By substituting these values of y — ?;, x — a, for yi — h, x-^ — a, 
respectively, in Eq. 2, we find 

dx' 
Eqs. 6 will dotormino the position of the centre: and Eq. 
7, the length of the radius of tlio osculatory circio io the 
given curve at any point. When the curve at the point of 



286 



DIFFERENTIAL CALCULUS. 



osculation is concave to the axis of x^ as is the case if y is 

positive and ~r^ negative, then, to make p positive, we must 

take the minus sign written before the second member of (7). 

The first of Eqs. 5 indicates that the centre of the circle is 

in the normal to the curve at the point of osculation ; and from 

the second of these equations we conclude that y — h and 

d'^y 

— 2 must have opposite signs, and hence that the centre of 

the circle is always on the concave side of the curve, since 
y — 6 is the difference between the ordinate of the point of 
contact and the ordinate of the centre of the osculatory circle. 
In general, the contact of an osculatory circle is of the 
second order, that is, of an even order ; and consequently it 
crosses the curve at the point of contact, except at particular 
points where the contact is of an order higher than the second. 
The osculatory circle is often called circle of curvature ; and 
its centre and radius, the centre and radius of curvature. 

172, As an application of the formulas of the preceding 
article, let it be required to find the radius of curvature of 
a conic section at any point of the curve. If the curve be 
referred to one of its axes, and to the tangent through its ver- 
tex, as co-ordinate axes, its equation 
will be 

y2 — 22DX -f- qoo'^, 
which, by two differentiations, gives 

dy^p-j-qx^ 2/— + (^— Y=^ 
dx y dx'^ \dx) 

In the last of these, substituting for 




dy 



dx 
have 



its value taken from the first, we 



RADIUS OF CURVATURE. 28' 

^d^'^ y' ~^' 



whence 







y^{\ 


dyy 




p. 



and for p we have 



The numerator of this value of p is the cube of the normal 
NN'; for from the triangle MNN' we have 

and p z= —- . 

Therefore the radius of curvature at any point of a conic sec- 
tion is equal to the cube of the normal at that point divided 
by the square of the semi-parameter. 

The value of p expressed in terms of the constants of the 
equation, and the abscissa of the point N, is 

p- — y 

17 3» The equation of the tangent lino to a curve at the 
point (^x^ y) being 

the expression for the length of the perpendicular p lot fall 
from the origin of co-ordinates on the tangent is 

dy 

JO 



X -/ — ?/ 

dx ^ 



J.T^' 



It 



288 DIFFERENTIAL CALCULUS. 

whence, by differentiation and reduction, 

dp _ dx' \ ^ \dx/ \ dx d x" \ dx ^ 

= ~A^ + yl] (Art. 171). 



dx ^ /^^'\ 2 ^ - 



i+mv " 



)l pV ' "^ dx 



^dx/ 

And, if r be the distance from the origin to the point of tan- 
gency, 

dij 
and, substituting this value o^ x -{-y-^ in the expression for 

-Y-, we have 
dx' 

dp 1 dr dr 

dx p dx^ ' ' dp' 

17 d. If X and y are both functions of a third variable, s, 
then 

dy d'^y dx d'^x dy 

dy ds d'^-y ds"^ ds ds'^ ds 

dx'^'dx' d^^~ /^Y 

ds \ds/ 

dt/ d (J 
and these values of ^^ , --^, put in the formula for p (Art. 

171), give 



/dxX"^ , fdyy } I 

(2). 



\dsn~ \ds 



d^y dx d'^x dy 
ds'^ ds ds^ ds 
Supposing s to be the arc of the curve estimated from a fixed 



RADIUS OF CURVATURE. 289 



point, we find, from the formula T" ^= ^ ^ + ( y ) ^^ ^^^- ^^^y 
ds~ L"T7M'' *** \dsl^\dx) \ds^ ~ 

!)■+ (!)■=' <»). 

1 1 _ d'^-y dx d'^x dy 

' d^y dx d^^x dy^ p ds'^ ds ds'^ ds 
ds^ ds ds^ ds 

From (3), by differentiating, we get 
dx d^ dyd2y__ 
ds ds' '^ ds ds' ^^^* 

Squaring (4) and (5), and adding, we find 
1 fd'x\'' . /d'ys^ 



p2 \ds'y \ds' J 

d IT d II 
Eliminating -— -, --j, in turn, between (4) and (5), observing 
as cLs" 

that 





©'+©■='. 


also find 






d'-y d^'x 

1 _ ds' _ d^' 

p~~ dx ~~ dy 

ds ds 



17S, To find the expression for the radius of ourvaturo in 
terms of the polar co-ordinates of the curve, we substitute 

in the value of /), Art. 171, the values of ; , , ',, jriven in 

dx dx' " 

Art. 131, thus getting 



290 DIFFERENTIAL CALCULUS. 



drV ^ ^ 



^ \dd) dd'' 
and, when r — ~, we have 

dr__ 1 du d'^r __ 2 /duV 1 d'^u 
dd~~u^dd' dd^~7? \dd) ~ u' dd^' 
and these values, substituted in the above value of p, give 

} \dO 



2 1 L^'^ 



176, The chord of curvature at any point of a curve is the 
portion of a secant line through that point that is included be- 
tween the point and the arc of the circle of curvature at the 
same point. 

The chord of curvature that, produced if necessary, passes 
through the pole, is obtained by multiplying 2p by the cosine 
of the angle included between the radius vector and the 
normal to the curve at the point ; but if r is the radius vec- 
tor, and p the perpendicular let fall from the pole on the 

tangent to the curve, - is the cosine of the angle included be- 
r 

tween the radius vector and the normal. But the value of p 
is readily found to be : hence the chord of cur- 

vature through the pole is equal to 

^'-r = ^P£=-77—^ (Arts. 173, 175). 



/duV 



RADIUS OF CURVATURE. 291 

177, Denoting by a the angle which the tangent to a curve 
at any point makes with the axis of abscissae, we have 

tan. a^=. -f . a == tan.~^-^: 
dx ax 

therefore 

dP'y cf^y 

da dx''- dx dx"^ 



dx 1 ' ds 

since -^ — — — — : therefore p =1 —- ■ 

^_j_/^V?i da 
\dx^ 

178, The co-ordinates of the points of the curve at which 
the radius of curvature is a maximum or a minimum must be 

found from the equation of the curve and the equation -^ :=0; 
the latter leading to 



K3)"£-gj-(l)'h" <"■ 

Differentiating the second of Eqs. 3, Art. 171, we find 

^d^,d^+^'^^^-^'^d^=''- 
^ 1 1 



d^y, dx. dx- \(!x- / dx 

' ' dxl Vx-b - ^.fdyV 

by Eqs. 4 of tlio same article. 

Comparing Ecjs. 1 and 'J, it is soon that -J^ - H; wliioh 

dx'^ dx'-' 
1 

proves, tliat, at tho points of niaxinuiiu or niiuiuunu our- 



292 



DIFFERENTIAL CALCULUS. 



vature, the osculatoiy circle has, with the given curve, a con- 
tact of the third order. 

179» If a perpendicular be let 
fall from the origin of co-ordinates 
on the tangents drawn to the differ- 
ent points of any curve, as S3IS\ the 
locus of the intersections of the per- 
pendiculars with the tangents will 
be a new curve, the properties of 
which will depend on those of the 
given curve. 

Denote the co-ordinates of the new curve by a?i, y^] then 
will the length of the perpendicular p^, from the origin to the 
tangent drawn to this curve at the point corresponding to the 
point (cc, y) of the given curve, have for its expression 




Pi 



^ dx]^ 



yi 



J 



1 + 



dxi) 



The equation of the tangent to the given curve is 

dy . , 

^ and V being the general co-ordinates. Since the point {x^, 
is on this tangent, 

''" X). 



y. 



y=%^^^ 



The equation of Oj) is (x = ^v; and, because Op isperpendic- 

Xi 

2/1 __ dx 



ular to Mp, 



or 



— -j- : whence 
^1 dy 

{yi-y)^i =—^i{^i — ^\ 

yyx^-xx^ — x\ -\-yK 



EVOLUTES OF PLANE CURVES. 293 

Differentiating this last with respect to x, we find 

y ^y A^ydy^^x A-x^'^'-^v ^y'4-2x.^-' 

Substituting for --^ its value. ^ , transposing and redu- 

dx ?/i 

cing, we have 

dy^ 2iCi — X ^ 

dx^~ ^i — y' 

and, by means of this, j9i becomes 

\/x^ -\- y^ ^ 

r being the distance from the origin to the point {x, y) of the 
given curve, andj9 the perpendicular Op let fall from the ori- 
gin on the tangent to the curve at the same point. 

180, If f{x, ?/) zrr be the equation of a curve, it has been 
shown (Art. 171), that, calling /*, r, the co-ordinates of the cen- 
tre of curvature corresponding to the point {x, y) of the given 
curve, we have 

^_p+(y_,)g = (1), 

'+(£)'+-"-'g=« <='■ 

By means of these equations, the otpiatiou of iho curve, and its 
first and second differential equations, wo mav eliminate .r. >i. 

-/; /.,, and find a direct relation btMwoon k and r. This will 
ax dx^ 

be the equation of a new curve, calliMl, with reference \o the 

given curve, ///o cvolu/c; the given i-nrve being flir inrohdc. 

It is evident that /< and »• mav be considered as iV.nctions i^f 



294 



DIFFERENTIAL CALCULUS. 



x; and, if Eq. 1 be differentiated under this supposition, we 



have 



1 + 






^dxj dx 

and, through (2), this reduces to 

dx dx dx ' • • 



d^i 
dx 



dv dy 
dx dx 



a/i ax 

dy 



whence, by the substitution of the value of ^ derived from 

dx 

the last of these equations, Eq. 1 becomes 



y 



dv . . 

V ^= -^ [x — ia). 



dfi 



These relations show that the tangent to the evolute is a nor- 
mal to the corresponding point of the involute, and the con- 
verse. 

A consequence of this property is, that the evolute of a 

curve is the locus of the 
intersections of the con- 
secutive normals to this 
curve. For take the two 
normals 3IK, If^j^^, which, 
by what precedes, are tan- 
gent to the evolute at the 
points K, K'. When the 
point M' is made to ap- 
proach the point if, the line M'K' approaches the line MK^ 
and the points K' and N tend to unite in the point K: hence 
the point ^may be regarded as the intersection of the normal 
JiXwith the normal indefinitely near or consecutive to it. 

Another important consequence is, that the length of the 
arc of the evolute between two centres of curvature is the 




EVOLUTES AND INVOLUTES. 295 

difference of the corresponding radii of curvature. To prove 
thisy differentiate the equation 

p' == (^ - i^Y -\-(y~ ^)\ 

treating y^ ^, v, and p as functions of x : we thus have 

dx \ dxj \dx dx/ 

which, by Eq. 1, reduces to 

dp . ^d\i dv . . 

From Eq. 1 and the equation ~ 4- -.^ — -^ = 0, we 2:et 

dx dx dx ^ 

when s denotes the length of the arc of the evolute estimated 
from any point F [Cor. 2, Art. 161) : whence 
, .da . .dv 

(a, _ ^.y (y - rf p dx ^"f- 

And, bv the combination of Eqs. a and c, we find -y- = =iz -^ : 

dx dx 

d{Szr:p) 

wherefore, since -^ = 0, it follows that 5 zpp is equal to 

some constant which Ave will denote by /; that is, 

5 zp p = ?, s^ zp /)^ = 7 ; . • . s — s' = p — p\ 
or arc FK' — arc FK — MK — Al'K' = arc ^A''. 

^ Suppose a flexible but inextensiblo string, of a length equal 
to M'K' plus the arc K'KF^ fastened by one of its ends \o F, 
to envelope the curve FKK', ami then pass in tlie direction of 
the tangent to the curve at K^ from K' to J/'. If this string 
be unwound from (his curve, its free end will describe the 



296 DIFFERENTIAL CALCULUS. 

curve MM'S, It is from this property that the terms " evolute " 
and " involute " are derived. It is also seen that there may be 
an unlimited number of involutes answering to the same evo- 
lute FKK'^ and that, to describe them, it is only necessary to 
lengthen or shorten arbitrarily the part of the string that ex- 
tends in the tangent to the e volute. Since the tangents to 
the e volute are normals to all these involutes, it follows that 
the latter curves have the same normals and the same centres 
of curvature, and that the parts of the common normals in- 
cluded between any two will be equal: hence one involute 
enables us to find all the others. 

1.81, Eadius of curvature and evolute of the ellipse. 

The equation of an ellipse, referred to its centre and axes, is 

1 dv V-x dP-y h^ 

whence -^ = ^^ ^, ~ ^— „• 

ax a^y ax^ a^y^ 

These values, substituted in the formula for the radius of 

curvature. Art. 171, give 

/ 5^\| 



a'V 



a'^y'^ 



To find the equation of the evolute of the ellipse, resume 
the equations 

^-^ + C2/-'')J = (1), 

■+(l)'+<»-'>3=; <^>. 

of Art. 180. Putting in Eq. 2, for -^ , -^, their values, it be- 
comes 

a^^ + Vx'^y - a'b\y — ?^) = : 



E VOLUTE OF THE ELLIPSE. 



297 



whence y — y = ^ ^ ' '^ _ ^ i \ 



«'^': 



a'b' 



y- 



Making a^ _ ^2 _ ^2^ ^^^ gj^j 

(5^ + c^?/^)y , c2?/3 c^y^ ,^, 

2/-^ = ^-^^-^ =-2/ + -^r- .-.--— /- (3). 

Substituting in (1) the value of y — ^ just found, we get, after 
reduction, and the elimination of y by means of the equation 

of the ellipse, jm = — ^- (4). 

Eq. 4 might have been derived from (3) by changing the sign 
of the latter, and in it writing x for y, and a for b. This is a 
consequence of the symmetry of the equation of the ellipse, 
and the relation between a, J, and c. 

a \m/ ' b \n/ 
Writing the equation of the ellipse under the form 



Put 



~ ^ m, — = n: then - — , , , , 
a a \m/ b 



+ 



= 1. 



X y 



this becomes, when the above values of -, ^ , are substituted. 



n 



+ -] =1 



for the equation of the e vo- 
lute. The form of this equa- 
tion shows tluvt the curve is 
symmetrical with respect to 
the axes of the ollij)se. For 
7- z= we have 

/( =: ± 9M = ziz . 

The curve has, iluM-c^tun*, two 
points, II, II', in the transverse 

88 




298 DIFFERENTIAL CALCULUS. 

axisj situated between the foci^ and equidistant from the cen- 

tre. Making ^u ziz 0, we find j^ = =b ti = rh — for the distances 

from the centre to the points E, E', at which the curve meets 
the conjugate axis. 

By two diiferentiations, we find 

m\m) n\n) d^ ^ 

m'^Km) n^\n/ \d^/ n\nj dfi^ 

whence 

d^v __ m^\m/ n\n) \d^ 

dp ^ ^, 

n\n 
Since the numerator of this expression is positive^ the sign of 

-,-^ will be the same as that of the denominator ; that is, -^-^ 
dfi^ ' aju- 

and 1^ will have the same sign. The evolute at all its points 
will therefore be convex towards the axis of x (Art. 155). 
Moreover, w^e have 



dv 


(:-) 


-i 


1 

m 

1 ~ 
n 


l^mv^i n 


d^- 


Q 


-i 


~ \)i{x/ m 



Since this differential co-efficient becomes sero for j^ = 0, and 
infinite for [^ =^0, we conclude that the axes of the ellipse are 
tangents to the evolute at the points H, H', and E, E' ; and 
that, in consequence of the symmetry of the curve with 
respect to the axes, these points are cusps. 

182* Radius of curvature, and evolute of the parabola. 

When referred to the principal vertex as the origin of co- 



CURVATURE AND EVOLUTE OF PARABOLA. 290 



ordlnates^ the equation of the parabola is y^ == 2pXj from which 
we find 

dy p d'^y p'^, 

dx y dx^- y'^ 

and, by means of these, the general value of p, Art. 171, be- 
comes, without respect to sign. 



9 — 



i+<^* 



y 



2\2 



P' 



p' 



To get the equation of the evolute, we must substitute the 

values of —^ -^, in 
dx dx^ 



i + (M+(.-")S=o, 



^dx 
which thus become 



dx 

d 
dx' 



^ — /^ + (2/ — ^) r = 0; 




The elimination of x and y betwoou these equations and the 
equation of the parabola leads to the equation of the ovoluto. 
From the second, we find 



U' ?/' //' 



r 



.•i' 



and, putting this value of r in tlie first, avo liave 



X — n -\-p -I- * - — ; . • . // —p ::^ r>.r. 



Thoroforo avo liavo 



300 



DIFFERENTIAL CALCULUS. 



jj2,|,^ x = -{li —29), y"- = 2j)x: 



whence ?/^ = ^'^i'^ 



y-=2^ 



{22jxy =z 



2 )3 



and 



I>'^' = 27^' ^^ ~^')'' ''^ "^ 27^ (^ ""-^')' 



for the required equation. 

If the origin of co-ordinates be transferred to a point at 
the distance p in the direction of positive abscissae, the new 
being parallel to the primitive axes, the equation of the evo- 
lute takes the form 



27j9 ^ ' 



or V 



rt 






"We readily recognize that this curve is symmetrical with re- 
spect to the axis of abscissse, and that it extends without limit 
in the direction of x positive. 
By differentiation, we find 

dv _S rj^ cV-r _3 \~T~ _i_ 1 1 

Therefore, at the origin of co-ordinates, the axis of x is tangent 
to the curve, and this point is a cusp ; and, since the sign of 

y-2 is the same as that of v^ the curve is at all points convex 
towards the axis of ic. 

183, The expression for 
the radius of curvature and 
the equation of the evolute 
of the hyperbola may be de- 
^ duced from those for the el- 
ii lipse by changing h'^- into 
— 5^. Thus we have, for the 
radius of curvature, 




CURVATURE AND E VOLUTE OF CYCLOID. 



301 



P = 



(h'x''-}-a^y')i 



and, for the equation of the evolute, 

after makina; c"^ z= a'^4-b'^, — = m, -j-:=zn. 

The form of this equation shows that the evolute of the hy- 
perbola is coraposed of two branches of unlimited extent, and 
symmetrical with respect to both axes of the hyperbola. It 
has two cusps situated on the transverse axis beyond the foci, 
and is convex at all points towards the transverse axis. 

ISd. Radius of curvature and evolute of the cycloid. 

du 

By squaring the value of — , which, for this curve, is 

ctoc 



__ |2r- 



(Art. 146), and differentiating, we find 



dy 

dx 



Substituting these values of -^- , ^yi ^^ ^^^^ general expres- 
sion for /), we have 



^dy d'^y __ 


2rdy^ 


. d'y 


r 


"^ dx dx^ 


y^ dx' 


• • dx'^ ~ 


t 



^r^y 



r- 



'.p=2V2ry. 



r 

Now, Fm = IN, and, 
from the right-angled 
triangle PNG, wo 
have 

PN^VGNxNf; 
that is, FN^ V'lry. 
Ilenco the radius 





a 






a' 


.^1 


c 


)i 


7 




^J^ 


NV 




^ 




OmT^ 


p4 


T 


\ 


D ^^ 


Vl/x 


/ 


\ 


(4 



302 ' DIFFERENTIAL CALCULUS. 

of curvature at any point of the cycloid is twice the normal 
at that point; and, if PN be produced until NQ = FN, the 
point Q will be the centre of curvature. 

18S. The property just demonstrated leads, by very sim- 
ple deductions, to the determination of the evolute of the 
cycloid. 

Produce the vertical diameter G^iVof the generating circle 
(figure last article), making JSfL == GN", and onNL, as a diame- 
ter, describe a circle. Through L, the lower extremity of this 
diameter, draw LE parallel to Ox, meeting the axis O'D, pro- 
duced in K The arcs PJSf, NQ, belonging to equal chords, 
are equal ; . • . arc NQ = ON: but OD = arc NQL ; . * . arc 
LQ = ND := LIJ. Thus it is seen, that if two equal cir- 
cles lying in the same plane be tangent to each other, and the 
one be rolled on the common tangent while the other is rolled 
on a parallel to it at the distance of the diameter of the circle, 
the points of the two circumferences which are common at the 
time of starting will, during the motion, generate two equal 
cycloids ; that generated by the point in the circumference of 
the second circle being the evolute of that generated by the 
point in the first. 

This relation between the two cycloids, generated as just 
described, may also be inferred from the property of the sup- 
plementary chords of the generating circle, which are drawn 
through the extremities of the vertical diameter of this circle 
in any of its positions, and the corresponding point of the cy- 
cloid (Art. 146). For, since FG is tangent to the cycloid 
OO'B at the point F, NQ, or Pi\^ produced, is tangent to the 
cycloid OQFJ at the point Q. Hence this last curve is the 
locus of the intersections of the consecutive normals to the cy- 
cloid OO'B, and is therefore its evolute. 



E VOLUTE OF THE CYCLOID. 303 

186, The application of the formulas of Art. 180 leads to 
the same result. 

From the equation of the cycloid, we have 

dy \2r—yd'^y r 

dx~~ \ y ' dx^ ~~ y'^' 

dij d II 
Substituting these values of —- , -— ^ , in the equations 

(aiX OjX 

^ - /^ + (2/ - ^) ^ =^ 0; 

'+(l)"+<^-)S=«. 

we find from the second 

2r r 

— _ (y _ r) — = 0, or 2ry — r{y — v) = 0; 

" if 

y=^-v (1); 

and from the first 

which, if we replace y by the value just found for it, and trans- 
pose, becomes 



x = ^J^2v^ ^- (2). 



V 



The equation of the cycloid 

X =: r COS. - 1 ^- — \/2ry — y'' (8\ 

by the substitution of these values of .r and //, becomes 

I 2r 4- V r 4- V 

fi _|_ 2v Z_ ^ r cos.-^ —^ V- 2r»' - r'' ( 4\ 

which is the equation of the evoluto. r>iit from Eqs. 1 and '1, 
it is seen, as it also is from (1), that there are no points of tlio 



304 



DIFFERENTIAL CALCULUS. 



curve for which v is positive. Making v negative, transposing 
and reducing, we have, finally, 
, r 



fi =z rcos. 



-^V2rv-v' (5), 

Now, let the reference of the 
curve be changed from Ox, Oy, 
to Ex^, Ey' ; positive abscisssB 
being estimated from E to- 
wards x'^ and positive ordi- 
nates from E towards y', x^ 
and yi denoting the new co-or- 
dinates of the evolute. Since 

, OD = Trr, DEzzz 2r, we have 

fi= OE- EI= nr - x„ v^ IF- FQ = 2r- y,. 

Eq. 5, by the substitution of these values of {a- and v, becomes 





p 


1 


•i 


1 






^ 


-y. 


-v<» 





^ 




^ 


D ^ 






1 


I 


' ] 


Q 



7tr 



X-, z= r cos." 



^^r-{2r-y,) 



J^s/2r{2r-2j,)-(2r-y^\ 



which reduces to 



Ttr — x-i = r cos, 



-i2/i-^ 



-\-V2ry, 



yl 



or 



rl 7t 



cos 



-i2/i 



— s/2ry^-y\. 



But COS.' 



-xV 



^=.n — COS." 



^ — 2/1. 



T r 

equation above, it becomes, finally, 

,r — 2/1 



introducing this in the 



X^:=r COS. 



V2ry, 



This equation difi'ers in no respect from (3), except in having 
Xi, yi, instead of x and y ; which shows that the evolute of a 
cycloid is an equal cycloid, situated, with reference to the axes 
Ex^, Ey'^ as the involute is with respect to the axes Ox, Oy. 
1S7» It has been proved (Art. 180) that the length of an 



ENVELOPES OF PLANE CURVES. ■ 305 

arc of the evolute to any curve is the difference of the radii 
of curvature corresponding to the extreme points of the arc. 
In the cycloid at the point (last figure)^ /> = 2 V^2r?/ == • 
hence PQ = 2FNF is the length of the arc OQ : and, with re- 
spect to the given cycloid, arc FO^ = 2PG. 

To express the arc PO^ in terms of the ordinate of the point 
P, we have 

arc PO' = 2PG = 2\/2r X GO. 
But GCz=^2r—y: ,' , ^rc PO' ^ ^VW''^^^^^. 

Making y =^ 0, in this value of PO^, we have arc O'O = 4r ; 
hence the entire arc of the cijcloid is four times the diameter of 
the generating circle. 

188. Envelopes. If one or more of the constants enter- 
ing the equation of a curve be changed in value, we shall have 
a new curve, differing in position and dimensions from the 
given curve, but agreeing with it in kind : that is, if the given 
curve be an ellipse, the new curve will be an ellipse ; if a pa- 
rabola, the new curve will be a parabola. The constants which 
thus change in value are called the variable parameters of the 
curve represented by the equation. 

The locus of the intersections, if any, of the consecutive 
curves of the same species, — that is, of curves whose equa- 
tions are derived from a given equation by causing one or 
more of its constants to vary by continuous deu'roes, — is 
called an cnvelojje. 

Suppose F{x, y,a) = (1) to ho the equation of a curve 
involving, among others, tlie constant a: and lot a he takou as 
the variable parameter. Changing a into a~\-Ji, tlie equation 
hocomci^ F[A\ //, a-\- /i) -~ ^'2), which reprosonts another 
curve belonging to the family of that represented bv ^\). 

o9 



306 DIFFERENTIAL CALCULUS. 

By Art. 56, Eq. 2 may be put under tlie form 

F{x,y,a)^hF^{x,y,a + dli) = () (3). 
Observing that F' signifies the derivative, with respect to a, 
of the function symbolized by F, Eqs. 1 and 3, when simulta- 
neous, are equivalent to 

F{x,y,a) = ^, F^{x,y,a^6li) = (4); 
and the values of x and ?/, determined by the combination of 
these equations, will be the co-ordinates of the intersection 
of the curves of which (1) and (3) are the equations. 

If h be diminished without limit, Eqs. 4 become 
F{x, y, a) = 0, F'{x, y, a) = (5) ; 
and the point determined by these equations is the limit of 
the intersections of the curves of which (1) and (2) are the 
equations. The equation which results from the elimination 
of a between Eqs. 5 will evidently be the envelope of the 
family of curves represented by the equation F{x, y, a) = 0, 
and of which the individual curves are formed by assigning 
different values to a. 

The envelope touches each curve of the series at the point 
common to the curve and the envelope. This is proved by 
showing that the envelope and the curve, at the common 
point, have the same tangent. 

Since (1) becomes the equation of the envelope when in it 
the value of a, deduced from the second of Eqs. 5, is substi- 
tuted, let (1) be differentiated under this supposition, treating 
X as the independent variable, and a as a function of x and y, 

and we have for finding the value of -j- for the envelope, 

dF clF dy dFi da da dy)__^ ,q^ 
'dx dy dx da \dx dy dx) 



ENVELOPES OF PLANE CURVES. 307 

But, at the point of intersection of the envelope with the given 
curve 

hence (6) reduces to 

dF dFd^i 

dx^ dy dx"^ ^^^' 
which is the same as that obtained by the differentiation of 
(1) : whence, at the common point, the tangent hne to the en- 
velope is also a tangent line to the given curve. 

Ex. 1. Find the envelope of the family of straight lines 

derived from the equation y =^ax-\ , by causing a to vary. 

Differentiating with respect to a, x and y being constant, 
we have 

'^ X 

2/ = i 2 \^mx, y'' = 4:mx : 
hence the envelope is a parabola. 

Ex. 2. Find the envelope of the straiglit lines represented 
by the equation y z::^ ax -\- (h'^a'^ -\- c')^, wlien a is made to 
vary. 

Differentiating with respect to a, we find 



m „ 

X ^ — : 

a^ 



= -^ + 



Ira ex 



Substituting this value of a in the given equation, wo liavo. 
after reduction, 

h' ^ c- ~ ^' 
whicli is the equation o^ an olli[)se referred to its centre and 
axes. 

In each oi^ the c^xainpU^s just given, it has been required to 
determine (he curve from the general c^^uatiou of the tangent 



308 DIFFERENTIAL CALCULUS. 

line. This process, being tlie inverse of that for finding the 
equation of the tangent line, is sometimes called 'Hhe inverse 
method of tangents." 

If a point be taken on the axis of a:? at a distance from the 
origin equal to m, and a line be drawn through this point, 

making, with the axis of x, an angle having for its tan- 

gent, the equation of this line is ?/ = [x — m), and it inter- 

sects the axis of y at the distance — from the origin. The 
equation of the perpendicular to this line, at its point of inter- 

111/ -T-I- 

section with the axis of y, is y =.ax -\ . Hence the geo- 

metrical interpretation of Ex. 1 is, " From a point in the axis 
of Xj at the distance m from the origin, draw lines intersect- 
ing the axis of y, and to these, at their points of intersection 
with the axis of y, draw perpendiculars ; required the enve- 
lope of these perpendiculars : '' and that of Ex. 2, ^' To find the 
envelope of a series of straight lines, so drawn that the product 
of the two ordinates of any one of these lines corresponding 
to the abscissas, -\-h^ — 5, shall be equal to c^." 

Ex. 3. Eind the envelope of all the parabolas given by the 

equation y =z ax — --^ — a;^, by causing a to vary. 

Differentiating with respect to «, we have 



\) =^x : , • . ft = 



P 



p x 

whence, by substituting this value of a in the given equation, 
we find, for the envelope, 

ic2 = 2pl-- y\ or x"- -|- l^jy -p = 0, 
which is the equation of a parabola. 



ENVELOPES OF PLANE CURVES. 309 

Ex. 4. Find the envelope of the normals drawn to the dif- 
ferent points of a given curve. 

Let the equation of the curve be y ^=:/(x) ; then the equar 
tion of the normal is 

«i-a' + (yi-2/)^^ = (1), 

in which x-^, yi, are the running co-ordinates of the normal. 

From the equation y z=/(^x), y and -^~ can be expressed in 

terms of x, and thus x becomes the variable parameter in 
Eq. 1. Hence the equation of the required envelope may be 
found by eliminating x between (1), and 

-'+*-") s-(iy=» «. 

which we get by differentiating (1) with respect to x. 

Comparing (1) and (2) with the formulas, Art. 171, it is 
seen that Xi, ^i^are the co-ordinates of the centre of curvature 
of the point (x, y) of the given curve ; that is, ilie envelope 
of the normals of a cui^ve is the cvolute of the curve. 

189, When the equation, representing the family of curves 
whose envelope is sought, involves several, say n variable pa- 
rameters, and these parameters are connected by )i — 1 inde- 
pendent equations, instead of effecting tlie elimination oi^ ri — I 
parameters, and then differentiating with respect to that which 
remains, we may proceed as follows: Let the ocjuation of tlie 

curve be 

F[x,y,a,h, c...) = ^\), 

and let the n — 1 ecpuitions of condition lor the paraniotor be 
fia,l>,c...).^.i) ^ ^^^^ 



/„_K(/, /'.('...") 



310 



DIFFERENTIAL CALCULUS. 



By reason of Eqs. 2, n — 1 of the parameters may be regarded 
as functions of the remaining one taken as independent. Let 
this be a, and differentiate Eqs. 1 and 2 with respect to it, 
thus getting 

dF dF db dF dc 

da'^'db da'^lTcda'^''"" ^'' 



dA ^ df 
da db 

da "^ db 


db df, 
da'^ dc 
db df, 
da^ dc 


dc 

da-^- 

dc 

da'^" 




da ^ db 


dh dA_, 
da~^ dc 


dc 

Ja'^ " 


• =:0 



(4). 



NoWj it is plain that if, in Eqs. 1 and 3, all the variable para- 
meters and their functions be expressed in terms of a, and a 
be then eliminated between these two equations, the resulting 
equation will be that of the envelope. To effect this elimina- 
tion, we have 2n equations ; viz., the n given equations, and 
their n differential equations : but there are only 2n — 1 quan- 
tities to eliminate; viz., the n quantities a,b,c..., and the 

n — 1 quantities -^-j -j- -- •' hence the ehmination is possible. 

Multiply the first of Eqs. 4 by the indeterminate l^, the sec- 
ond by I,, and so on, and add the results and Eq. 3 together: 
we thus get 



da da " da 

, A^^ , ; dA df 



+ '■ 



+ 



dF dfi 



db 
'dc 



-}-'" + 



' da 

dfn-i\db_ 
db ) da 
dA_^ clc^ 
da 



dc 



(5). 



ENVELOPES OF PLANE CURVES. 



311 



By means of the n — 1 indeterminate multipliers l^, l^. • • •? "^-n-ii 
we may satisfy n — 1 conditions. Let these be that the co-ef- 
ficients of ^ ,-,-..., in Eq. 5, shall reduce to zero. These, 

ClCi (jjQj 

together with that expressed by Eq. 5 itself, lead to 



dF df, df, 

da ^ da ^ da 



dF 



^ da 

d.A 



db^^'dE^^'di'^ 

dF ^df, ^ ^ df. 



4_3 ^"-^ — 



dc 



+ 



dc '^ do 



+ K. 



da 



dc 



\ (6). 



= 



We have now the 2n — 1 quantities a,h,c..., l^y ^-i- • •? ^^n-n 
to eliminate between the 2/1 equations (1), (2), and (6); and 
the result, being an equation between x and y only, will be the 
equation of the required envelope. 

190. When the general equation of the family of curves 
contains only two variable parameters, and they are connected 
by one equation, the process admits of the simplification, and 
the result takes a form the same as those in Art. 128. 

Ex. 1. Find the envelope to the different positions of a 
straight lino of a given length extending from the axis of .r io 
the axis of y. 

Let c bo the length of the lino, and a and h be the inter- 
cepts on the axes of x and y respectively ; then the oipiatiou 
of the line is 

and the ocpiation connecting a ;nul h is 
(f-' + Z/'^c- ^2). 



312 



DIFFERENTIAL CALCULUS. 



Differentiating (1) and (2) with respect to a and h, a being 
taken as independent, we have 

X y db dh 

and therefore, according to Art. 128, 
X y X y 

d^- h^ a h 1 

whence a =: icscf, h = y^c^, and 

a"^ -\- h'^ zzz c'^ z=z (.xt -j- ya) c3 : ,' , x^ -{- yi ^^ c^ 

is the equation of the envelope. The 
figure represents the curve traced in 
the several angles of the co-ordinate 
axes. 

Ex. 2. Find the envelope of the se- 
ries of ellipses formed bj varying a 
and h in the equation 

9 9 

a'-^ b ~ ' 
a and h being subject to the condition ab i= c'-. By differen- 
tiating with respect to a and b, regarding a as independent, 
we have 




T5 + 



a' 



vim 

b'^ da 



= 0, - 
a 



1 1 db 



b da 



= 0; 



whence 



g^lJ^i; r,a = xs/^, b = yV2: 



^y = ^: 



which is the equation of an hyperbola referred to its centre, 
and asymptotes as axes. 



EXAMPLES. 513 

EXAMPLES. 

1. What is the radius of curvature of the curve 

y ^z x^ — 4:X^ — ISx^ 

at the origin of co-ordinates ? . 1 

Ans. p = — 

2. Find the parabola which has the most intimate contact 
with the curve y t=: -^ at the point having a for its abscissa, 

(X 

the axis of the parabola being parallel to the axis of?/. 

3. Show that, at one of the points where y =: in the curve 



p'= 



ax (x — 3a) 
X — ia 



the radius of curvature is - ; and at the other, . 

4. What is the radius of curvature of the spiral of Archime- 
des, the polar equation of this spiral being r := aO ? 

Ans. /> = ± — -— — -, 
la--\-r- 

5. The Lcmniscata of Bernoulli is the locus of the points in 
which the tangents at the diilercnt points of an equilateral 
hyperbola are intersected by tlie perpendiculars let fall upon 
them from the centre of the hyperbola. Its polar equation is 
r'^ = a^ COS. 2^>/. What are the radius of curvature and the 
chord of curvature at any point of this curve? 

Ans. /) = ., ; chord of curvature ^= " r. 
or 3 

(). If a curve have // :— ^ (<"• -|- <' ■ ) f^^i' i^^ Oipiation. ]u-ove 

that the general co-ordinates ofils contrc^ o^cur^at^ro are 

x^ = x— ij ^^.. - I, y,~'lii. 

4 
40 



314 DIFFERENTIAL CALCULUS. 

7. What is the envelope of all ellipses having a constant 
area, the axes being coincident? 

Ans. 4^-?/^ = c*; nc"- being the given area. 

8. Find the envelope of the curves represented by the equa- 
tion 



h J ' \ k 

a and h being the variable parameters connected by the equa- 
tion 

^°^-A^ + F = *- 

9. Find the envelope of the system of straight lines con- 
necting, pair by pair, the feet of the perpendiculars let fall 
from the different points of an ellipse upon its axes ; the equa- 
tion of the ellipse being 

-- -|- ^ — 1. 

10. What is the envelope of the series of circles, the circum- 
ferences of which pass through the origin, and which have 
their centres on the curve of which the equation is 

a'^y'' -h^2ax — x'') i= ? 

Ans. {x' + y' — 2axf - ia'x'^ - Ab'-y' z= 0. . 



INTEGRAL CALCULUS. 



.<=x^~ 



SECTIO:^ I. 



MEANING OF INTEGRATION. — NOTATION. — DEFINITE AND INDEFI- 
NITE INTEGRALS. — DIRECT INTEGRATION OF EXPLICIT FUNC- 
TIONS OF A SINGLE VARIABLE. — INTEGRATION OF A SUM. — 
INTEGRATION BY PARTS. — BY SUBSTITUTION. 



101, Any given function of a single variable may always 
be regarded as the differential co-efficient of some other func- 
tion of the same variable ; that is, there is some second func- 
tion, which, when differentiated, will have the given function 
for its differential co-efficient. 

For ]ct/{x) be the given function. If this admits of possi- 
ble values for real values of x, 
we may construct the curve 
CPD, which, referred to the 
rectangular axes Ox, Oy, has 
y —f{x) for its equation. Tho 
area included between this 
curve and tlie axis of .r, that is 
limited on the one side by the fixed ordinate CJ, correspond- 
ing to x — a, and on tho other by (ho ordinate P.V. corre- 
sponding to tho variable abscissa x, is evidently a I'unction of 



V 






p^ 


)^ 




( 
/ 


Y 











A 


J!l 


I M' 




t 











316 INTEGRAL CALCULUS. 

x; and, of this function, y oy/(x) is the differential co-efficient 
(Art. 164) : hence we should have 

■^{areiiACPM)=:/{x), 
ctx 

Y (area A CPM) dx =/{x) dx. 

192* It will be found that the operations of the Integral 
Calculus are mainly those of passing from given functions to 
others, which, by differentiation, would produce the given 
functions. The fact that these operations are the inverse of 
those of the Differential Calculus has been taken as the basis 
of the definition of the Integral Calculus. But the fundamen- 
tal proposition of the Integral Calculus is the summation of a 
certain infinite series of infinitely small terms. To effect this 
summation, we must generally know the function of Avhich a 
given function is the differential co-efficient. The proposition 
may be stated thus : — 

Let f{x) be a function of x, which is finite and continuous 
for all values of x between x^, ic„, and of invariable sign be- 
tween these limits. Let cc„ be greater than x^^, and divide the 
difference x.^ — x^ into a number oi of parts, equal or unequal, 
represented by cc^ — X(^, x^ — ^i, x^—x^...^) x^ — Xn_i] re- 
quired the sum of the series 

S—f{x^) (^i-^o)+/(^l)(^2 — ^'l)-| 

when the number of parts into which x^ — x^^ is divided is in 
creased without limit, or n is made infinite. For brevity, 
denote the intervals x^ — Xq, x.2 — x^. , ., x^ — ^«_i; by 
h^, Ji2' • ', hnj and the series becomes 

S=/{X,)h,^/{x,)h,+ "■ +A^n-d^K-l+AXn-l)K (1). 



MEANING OF INTEGRATION. 317 

Now suppose F{x) to be the function of cc^ of which /(cc) is 
the first derived function ; then 

But, before passing to the limit, we should have 

^(^^+-'1-:^^^ =/(.)+. (AH. 15), 

p being a quantity that vanishes with h : therefore 

F(^x + h)-F{x) = li\f{x)+p\ (2). 

In (2), giving to h the values h^, li.^..., li^, and to x the 
values o^o, Xi...j x^_^^ x^^, and denoting the corresponding 
values of p by Pi, p2- - •, Pm observing that 

'^o ~r '^ 1 -— ■ "^ 1 > '^ 1 ~r '^2 — - ^2 • • • ; 
we have 

F{x,)-F{x,):=.h,\f{x,)+P,\, 

F{x,) - F{x,) =h,\f{x,)+P,\, 



F{x,_,) - F{x,,_,) = h,,_, {/(x„_,) + p„_i j , 
F{x„) - F(x„_,) = A, \f{x,_,) + p, j . 

Adding these equations member to member, for the first 
member of the result, we have F{x„) — F{x^)). The second 
member is composed of two scries, the terms of one being of 
the form Ji/{x) ; and of the other, Jip. Denote the sum oi^ the 
terms of the first series W ^/{x)Ji, and of the second by 2.pli : 
then our result may bo written 

F{x,,) - F{x,) = l/Xx) h + Ink (3). 
Jf p', the greatest among the quantities />, . p._, . . . , p,. . be sub- 
stituted for p in tlie series represented by 2/)//, we should have 
^Ph <P\J>i + Ii,+ ---+ /^,) -- P\x^ - X,). 



318 INTEGRAL CALCULUS. 

But p' vanishes wlien li is decreased without limit : hence 

F(x„) - F[x,) 
is the value towaixls ichich the series Jf(x)h converges when 
the quantities of ichich h is the type are diminished without 
limit; that is , 

lim.^/(x)A = F(x„)-i^(a)„) (4). 

193* It may be readily proved that If{x)h has a definite 
value when h is indefinitely decreased, and when, therefore, 
the number of parts into which the interval x^^ —x^ is divided 
becomes infinite. For let A^^ be the least, and Ai the greatest, 
of the values assumed hj /(x) for values oi x between cCo> ^«' 
then 

^f{x)h yA^{h^^h^-\ ^h^) —A^{x^ — x^), 

2/{x)h<:;^A,{h,^h^-\ ^h„) ^A,{x^-x,)', 

and since, by hypothesis, both Aq and Ai are finite, the same 
is true of 2f{x)h. It is evident that the values of/{x) inter- 
mediate to ^oj -^17 ^^ill ^6 furnished by the expression 

/\xo + d{x^-x,)\, 

6 being a proper fraction ; and that such a value can be as- 
signed to as will make 

^f{x)h = (x^ - x,)/\ x,-\-d (x,, — x,) \ 

a true equation. 

194. Putting Eq. 3 of Art. 192 under the form 

2/(x)h = F(x,) - F{x,) - 2ph, 

it is seen that the value of J^f(x)h will, in general, depend on 
the number and value of the parts hi, 7^2 . . ., A„, into which 
the interval Xj^ — Xq is divided, but that ]im.2/[x)h, for which 
^ph vanishes, is independent of the mode of division. When 



DEFINITE AND INDEFINITE INTEGRALS. 319 

all the parts into Avliich x^ — x^^ is divided are equal, each is 
equal to -^ " ; and any one of the intermediate values of x, 

T 

as x^^ is equal to x^-\ — {x^ — ic„). In this case, the value of 
\mi.2^f{x)li is represented by / ''f{x)dx^^F{x^^ — F{x^. 
The symbol T signifies sum.^ and dx represents the llz=l^x of 
the expression 2Lf(^x)h. The quantity 

rj\x)dx = F{x,)-F{x,) 
is called a definite integral ; the operation by Avhich we pass 
f{x)dxis called integration; and x^^ Xf^, 

are the limits of the integral. Since F{x^^) — F(x^) is the 
value of this definite integral, we must first find the function 
F{x) of X, of which /{x) is the difierential co-efiicicnt. The 
relation between /(a?) and F(x) is expressed by 

which, by the notation of the Integral Calculus, is 
f/{x) dx =z F{x). 

195, The function F{x) of x^ which, difi'erentiated, would 
reproduce f{x) dx, is denominated indefinite integral. But 
a constant connected with a function by the sign plus or mi- 
lius disappears in differentiation ; therefore the more general 
relation l)etweeny(.r) and F(x) is 

f/{x) dx = F[x) ± C: 
so that the proper value of ^ to verify (be equation 

df/ — /(.t) dx 
is given by the eipialion 



320 INTEGRAL CALCULUS. 

and the two symbols d and /, the one indicating differentia- 
tion, and the other integration, neutralize each other, and we 
shall always have 

Jdu ^:.u z^ C, djdu := du. 

The constant thus added to an indefinite integral is called 
the arbitrary constant of integration, or, simply, the arbitrary 
constant; it being any quantity which does not depend on the 
independent variable x. 

The operation of passing from an indefinite integral to a 
definite integral consists in substituting in the indefinite, suc- 
cessively, the limiting values of the independent variable, and 
taking the difference of the results. The arbitrary constant 
will, of course, disappear in the subtraction. 

196, In differentiation, constant factors may be written 
before the sign of differentiation. The same may be done in 
integration. For 

fdau = au. a Jdu = au ; 

Jdau = a Jdu, or fadu = a Jdu ; 
or, more generally, 

faf(x) dx = aj/(x) dx. 

f[x)dx is the expression for the limit of 

the sum I:f{x)^x, that is, the expression for this sum, when 
the number of parts of the interval x^ — x^ is increased with- 
out limit, and the value of the parts severally correspondingly 
decreased, it is evident, that, at the limit, the addition or omis- 
sion of a finite number of the components y (a?) A oj of lf{x)^x 
would not affect the result. 



DIRECT INTEGRATION. 321 

A single term /(a:^) a a? of the expression -^(cc) a a:; is called an 
element. 

197» Direct integration of simple functions. 

We sliall, for the present, confine ourselves to the deter- 
mination of indefinite integrals, to which it must be understood 
that an arbitrary constant is to be added. 

There are many cases in which a function is at once recog- 
nized to be the differential co-efficient of another. In such 
cases, we have simply to write the second as the integral of 
the first. 

Subjoined is a table of the integrals of the simple functions. 



fx^dx = — ; — -, fa^ dx ^= , , 

fsin. xdx = — COS. x^ fe^dx =: e^, 

fcos.xdx := sin. ic, / - = Ix, 

*J X 



/dx [* dx . 1 ^ 1 ^ 
17— = tan. a;, / — 7=7 ^ 5 = sin.~^ = — cos.~ --, 
cos.^'ic ' •^ V<^ — ^ <^ ^ 

/dx r dx . 

-. — r~ = — cot. X, f . = sni.~^ ic = — cos.~^ X, 

^m.-x ' 'J V 1 — X- 



r dx 1 ^ .X 1 
/ , , -^2 = - tan.-^ — 



= cot~^ -, 



a a 

In all of these formulas, x may be the independent variabk\ 

or it may be any function of the independent variable ; for if, 

r a-" + ^ 

m the formula J J?" (/.^* = - -j— r, '^' be replaced by f{x), we 

should have 

f\fix)\"dAx)^^^^_^^. 

r" + ^ 
Wlieu n = — 1, the formula I .v"dx — ' ■ reduces to 

•^ n -\- 1 



41 



rdx 1 



322 INTEGRAL CALCULUS. 

whereas, we know that I — ^=zlx. The failure of the formula 

t/ OG 

to give the true result in this case arises from the fact that 
the transcendental quantity Ix cannot be represented by an 
algebraic expression. It may, however, by a suitable trans- 
formation, be made to give the true value of fx"dx when 



/^+i 



71 =:r — 1. Take the general formula Jx^dx = ^ -f" ^j 

which may be written 

^ n-\-l Ti+l'n + l' 

Now, the term A --z, , in the second member, may be in- 

eluded in the arbitrary constant C : and thus w^e have 

//y.» + l 1 /y.tt + 1 1 

n-^l n+1 n-{-l ' 

or, omitting the constant, 

x"dx z= , ^ — zzz - when 72 zzz — 1. 

»' n-\-l 

The true value ot this is found by differentiating the nume- 
rator and denominator with respect to n, and taking the ratio 
of the differential co-efficients (Art. 101). We find 



=zlx. 
n-\-l /n=-l\ 1 /«=-l 

198, The rules of the Differential Calculus enable us to 
find the differential co-efficients of all known functions ; but 
the inverse operation, of deducing the function of which a 
given function is the differential co-efficient, is not always pos- 
sible. Whatever the assumed function may be, there must be 
some other function of the quantities involved, which, differ- 
entiated, would produce it (Art. 191). The Second of the two 
functions thus related as differential to integral may not be- 



INTEGRATION OF A SUM. 323 

long to any of the small class of simple functions which have 
been admitted into analysis, or to any combination of such 
functions ; in which case, we are limited to series and approxi- 
mations for the expression of integrals. For example, we rec- 

1 X 

offnize , — — - to be the differential co-efficient of sin.~^- , 

or that / / ., — - =z sin.~^ -, because the latter function has 

been named, and its properties investigated. Had this not 

/dx 
—, — , could not have been ex- 

V« — ^ 

pressed by means of a simple function. 

199. Integration of a sum of functions of the same vari- 
able. 

In the Differential Calculus (Art. 19), it is proved that if 

y =/{-Tc) ± 9 (^) ± V^ (^) =h . . . , 

or df/ =/' (x) dx rb (p^x) dx ± i// (x) dx . . .: 

whence 

fdi/ ~y=- Jf{x)dx ± f(p'{x)dx i fw'{x)dx .... 

Hence the integral of the sum of any number of functions is 
the sum of the integrals of the component functions. For ex- 
ample, 

r Ai:"'+^ 7>V+^ Cxf'+^ 

also 

J(r).r'«- 7.r'^+ -l.r — ^)dx = x' - J .r"* -|- ±v' - .V, 

and j(^-^f^ dx = I .r' - ±r'^ 4- o^r. 



324 INTEGRAL CALCULUS. 

200, Integration by parts. 

If u and V are functions of the same variable, we have, by 

differentiation, 

d{uv) _ dv du 
dx dx dx' 

The integration of both members of this gives 

r dv y , r du 7 
uv = u-^ dx 4- V -r- dx: 

^ dx '^ dx 

therefore 

f. dv . ^ du . 

lo-z- dx =1 uv — V—- dx, 
^ dx '^ dx 

or Judv =1 uv —Jvdu. 

This method of integration, by which the determination of 
an integral Judv is reduced to that of another fvdu, is fre- 
quently employed, and is called integration hy jjarts. 

Ex. 1. fx"^ COS. xdx. 

Put x''^ z= u, COS. xdx = dv =^ d sin. x ; then, by the formula, 
fx"^ COS. xdx = Jx'^-d sin. x = x"^ sin. x — 2fx sin. xdx, 

Jx sin. xdx = — Jxd cos. x = — x cos. ^ -\- f cos. xdx 

= — X COS. X -{- sin. X. - 
We shall therefore have, by the substitution of this value in 
the first integral, 

Jx"^ cos. xdx = cc^ sin. x -]- 2x cos. x — 2 sin. x. 
Ex. 2. Jlxdx. 

Make Za? = w, c?jc = dv ; then Jlxdx = xlx — x. 
Ex. 3. Jx^'e^'dx. 

Making x"' = i^, e'^dx = c^v, the formula gives 
Tic^e^cZx = ic"e^ — njx''~^e'^dx, 



INTEGRATION BY SUBSTITUTION 325 

and the integration of x^e'^dx is thus brought to that of 
x"'~^e^dx. By another application of the formula, the expres- 
sion to be integrated would become x^~'^e^dx; so that, if n be 
a positive whole number, the proposition would be reduced, 
after n applications of the formula, to finding the integral 
e'^dx = de^. Hence, by a series of substitutions, we should 
have the required integral. 

Making ?i := 1, fxe^dx = e^(x— 1), 

« n — 2, fx^e'^dx = x'^e'' — 2fxe'^dx 

= e^(a;2- 2ic + 2). 

201, Integration by substitution. 

It is sometimes the case that a differential expression, 
f(^x)dx, which is not immediately integrable, becomes so by 
replacing the independent variable by some function of a new 
variable. The function selected must be such that it shall be 
capable of assuming all the values of the variable for which it 
is substituted within the assigned limits of the integral. 

Let t be the new variable, and suppose x =■ (p (t) ; then, by 
the Differential Calculus, 

dx 

^==c/(0, or dx^cp^{t)dt, 

and f{x)dx=f{cp(t)}if^{t)dt: 

whence, by integration, 

in which it must be remembered, that, if the first intoirral is to 
bo taken between the limits a and ?>, the second is to bo taken 
between the corresponding limits a' and h' . 

Ex. 1. j^ax^iydx. 



326 INTEGRAL CALCULUS. 



Put ax -\-h ^=^t, whence dx^:^-dt ; and therefore 

a 



J {ax + hfdx = ^fVdt = 



1 t^ + - 



a^ a n -\- 1 

Replacing t hy its value, we have, for the required integral, 



/ (ax + oYdx = - ^ ' — ^— , 



+ 

Ex. 2. r^x^d^ 

J^x'^^ 

Make 8:c^ + 5 = ^, then Sic^c?^ = 1 c^^J, 

8 

, c Sx^dx rl dt 1 ,, 

therefore /|^^ = J /(8.' + 5). 

It is evident from these two examples that success in efifect- 
ing integration by substitution must depend on the ingenuity 
of the student, and his knowledge of the forms of the differ- 
entials of the simple functions. 

Miscellaneous Examples. 

/xdx 

— , — - ' Make Va'^ -^ x'^ = t : . • . a^ _|_ ^2 _ ^2^ 
Va + x^ 

xdx = tdt ; and therefore 

r xdx . , 

2. JVa'-x'^dx, 



Putting Va^ — x'^ = u^ xz=^v, and integrating by parts 
(Art. 200), we have 

/Va^ - a;2 dx = x^-a^Zr^2j^ r x^dx^ 



MISCELLANEOUS EXAMPLES. 327 

/» /^ 2 /V.2 

But / V«''^ — ^'^ dx = I /- - o?a? 






Therefore, by the addition of (1) and (2), 

2 fVa' — x'' dx — x\/cC' - a;^ , ^2 f <^^ . 
and since 

a^r ,^1 z^a^sin-ig, Art. 197, 

we have finally 

r>s/^2Tr^2 (^^. ^ 1 a,y'^2-ir^_^ 1 ^2 gin -1 ^. 



dx 

=-. MakeVcc"'- 

if — a? 






Hence, by differentiation, dx = — 7 — dt : therefore 

J y^^2 _|_ti2 J t— X t J t 

^l{x^\/x''-\-a'). 



dx 



4^ I -— :,,, ^ Making V.^'- — a'- =z t — x, and proceed- 

ing as in Ex. 3, we should find 

•^ V aj-^ — «- 

5. /Va;2~+ tt- (/.r. Integrating by parts, Art. 1200. wo liavo 



328 INTEGRAL CALCULUS. 

But /Va^i=^ dx =z r^ !+^ ^^ 

Therefore, by the addition of(l)and(2), 

2fVx'' + «2 cZic =r ccVic^ + a' + a' r ^"^ 



By Ex. 3, 

'^ V iz;^ + ^ 
and hence 

J \/x'^ -\-a^ dx = -x\/oc^ + a^ + -^ l{x -i- Vx^ -\- a'). 

6. /Va^^ _ ^2 J^ _ _ ^^^^2 _ 0^2 _ _ ^^^ _|_ ^^2 _ ^2)^ 

. ^ The quantity under the radical si^n 

yx^-{-px-\-q 

in the denominator may be put under the form 

hence 

r dx /» dx 

Making x-^^~t, and q — ^z=a\ we have dx — dt, and 

IvFT7' = ^(^ + ^^'H^') V Ex. 3. 
In this last, substituting for t and a their values, we have 



MISCELLANEOUS EXAMPLES. 



I 



dx 



\/ic^ -\- jpx -\- q 



='('+i+J(-+i)"+»-'i 



l[x-\--^-]-s/x' ^px-^q 



8. Js/x'^px^qdx^j\i^-\-^^ -{-q-^V dx. 
Put X -\--^=it, ^ — ^ = <^^ then 
/V^HKpM^ ^^ ~ fVi^'+aJ dt 

by Ex. 5. Replacing t and a^ by their values, we have, finally, 
fVx'' -{-jox + q dx=i-fx+^ \^x' +px + q 



P' 



P 



«• s. 



dx 



\/2ax — x'^ 
and dx =^ — dt : therefore 

dx n dt 



Let ic = a — t; then 2aj:; — cc' 1= a- — / ', 



r ax n dt , t 

/ /,v ., — — / ^-^-—1 = cos -^ - 

*^ y Zax — x^ Js/a-' — t- ct 



= COS." 



a — X 



= vers." 



dx 



10. r^j*^^ 



-• Put x= -,--,; then 



2ax — «2 — v_ 1 _/ xv2ax — a-) = - — — V-^LJ_\ 
1 — ^ ^ 1 ~t\ l—t ' 



7 ^^^^ ^ ^^ 

ax = — ., ; a 1 u i t h re 1 re 

(I t)' 



/» 



/ dx _ 

x^2ax-^'~ 



adt 



ty 







(1- 


ty 


1 


a 


(7 


\\ + f) 
1 - r 



__ 1 r dt 

~ aJ s\'-t^ 



42 



330 INTEGRAL CALCULUS. 

a a X 

11. fx COS. axdx. Assume u = -, v ~ sin.ax ; whence 

dv — a COS. axdx and Jx cos. axdx — fudv: therefore 

r J X sin. ax r sin. ax , 

/ X cos. axdx = I dx 

'' a J a 

xsm.ax cos. ax 



a a 

sin. ax 



-7 v:=^e' 



12. J e*^^ sin. axdx. Put u 

r . -, sin.a.T ,^ me^'^cos.aa:; , 
e""^ sm. axdx := e""^ — / —ax. 

But we have, in like manner, 

/ae'^^GOs.ax , acos.aa; ^^ , ra^sin.aa? ^^ , 
- — dx = 2 — ^ + / 2— -^ ^^ * 

hence 

r . , sin.aa? ^^ acos.ao; ,^ m^ sin. ax ^^ , 
Je'^'sin.axdx = — e'^^ 2 — ^ ~ j ' 2 — e'^'dx, 

which, by transposition and reduction, becomes 

r „^ . 7 e^'-^fcsin.ax — a cos. ax) 

/ e'^^ sm. axdx ^ — ^ 5—, — ^ • 

^ a^-\-c^ 

13. fe''^ COS. axdx. 

Proceeding with this as with the last example, we should 

find 

r ^ e'^^fccos.ax 4- asin. ax) 

/ e*"^ COS. axax = — ^ o—, — 2 • 

J a- -\- c^ 

c ^-^ _ r ^^ 

*f\/(o-\^vx — x^) *' 



i'~r~2 



Put a + V = <^^ X —^=zt : . • . dx — dt 

^4 2 



MISCELLANEOUS EXAMPLES, 331 

Therefore 

f ^'^ ^ f "^1 ^ sin -1 ^ 



I 



X — ny, rt^ 

= sin,~^ ^ sin.~^ -^ 



15. /V(ff +P^ - ^')dx =/J W^ - (^ -|JJ ^*- 
Making g -f^ = a^, aj — ^ = ^, we have 



Jy/{q +px— x'')dx=fVa'' — t^ dt 

^ ^ V^^^=^ + ^ sin -1 - , by Ex. 2, 

1 , 1 Ox r> 

= - (2x - ^)v^4g +1^-^ + g (42 + j.^) sin -1 ^4^^ 

/^^ -r. 1 1 7 dt 
— . - . rut X =z -: whence dx := -, and 

ccV ^ — ct" t t 

c?:z; r (7^ 1 dt 



/ax r 

cr.K/ T."^ — a^ J 



\ X^ — 0} *^ Vl 



H'' a \\_ 

\a^ 



== sin.~' at ^= sin.~^ - • 

a a. X 



But, since 



- a . a Tt 1 . , .r 1 a \ rr 

Bin — [-cos.~^- = -, sin.~ --^ cos.~^ -: 

X X 2 a ii a x a - 

hence, throwing into the constant ot mtogralion, wo 

may write 

/dx 1 , (7 

— 7 o ., = - cosr^ — 
x\x^ — a^ a x 



332 INTEGRAL CALCULUS. 

17. f , =^' Makea;=-: ,' , dx ^=. x, and 

r ^^ if dt 17/,, ITT^ 



hf^ , |1^1\_ ly^ + V^^ 



ct \x yix^ a\l a 



X' 



ax 



11 X 1 X 



a .a a + Va^iaj^ a a + Va^ioj'^ 
by including - ?a in the constant of integration. 

Cli 

•^ ic^ — a^ 2a •^ \a? — a x -\- a/ 
1 r dx 1 r dx 

2a*^ X — a 2a^x-]-a 

1 7/ ^ 1 7/ N 1 7 ^ — ^ 

2a ^ ' 2a ^ ^ ^ 2a a; + a 

If X is less than a, then 

r dx r dx 1 r/ dx dx 

^ x'^ — a^ *^ a^ — x'^ 2a^ \a — x a 



1 7 N 1 7 N 1 J a — X 

^ — l(a — x) — -—l(a -\- x) z= —- I — ; — : 
2a ^ ^ 2a ^ ^ ^ 2a a + a?' 



and 




19. 



p 
Suppose q — to be negative, and make 



MISCELLANEOUS EXAMPLES. 333 



then, by the last example, 

/dx r dt ^ 

nf,/2 I ^n,. I ^y »/ /■- r/,'^ 



dx r dt 1 7 ^ — ^ 

cc^ -[-px -\- q "^ t' — a^ 2a t -\- a 



1 2x -{- p — \/ 4tq — p' 



V4:q—p'^ 2x -\- p -{- \/ 4:q — p' 

p P" 

If g — 7" ^^ positive, then q — — =: a^ ; and 

dx _ r ^^ __ ^ + -1 ^ 



/dx /* 

i^;''^ 4- z>a3 4- r7 J \ 



X- -\- px -\- q J t^ -^ a^ 



= - tan." 



tan. 



.1 2x +p 
^/\q—p^ ^/^q — p^ 

I , mp , 7np 

20. f-J!pl^dx = ,i ^ ^ d. 

J x^ -\-px -\- q I x^ -\-px -\- q 



2 •^ fl:;2 _j_ p^ _|_ g \^ 2 /'^ x^ -\-px-{- q 

The integral of the first term is -^ I{x'^ -{-px -f- q), and that 



of the second is found by Ex. 19. 

r dx rsiu.x , r dcos.x 

21. J -. =J -.-r-dx = —J -. 

^ sm. X ^ sin.-U/' *^ 1 — COS.- a; 

Make cos. x =.t; then 

by Ex. 18 : but 

1,1-4- COS. X 7 /I — cos. x\k , , .r 

^ = M , .— I == I tan. - • 



2 1 — cos. X \1 + COS. X. 

dx r COS. xdx /* (7 sin. .t* 



r (tx^ _ /* COS. xdx _ /* (/ sui. .t* 
J COS. X J COS," a; J 1 — sin."^ a; 



334 INTEGRAL CALCULUS. 

77 (j :j ' — i . 

A 1 — sm. X I ][ 

cos.-ic — sin.-ic 

23. f^^ = f'J^^^±^°^dx 

J sm. X COS. X J sm. a; cos. x 

= r(tan. cc + cot. x)dx 

= — Zcos.a; + Zsin.ir=:Z ~ :=Ztan.a;. 

cos. a; 

c?a? r sin.^ a^ + cos.^ x 



24. r . /^ , =r ^'°--f+°°f^ cza. 

J sm.'' a; cos.'' a? J sm/ x cos.'' ic 



25. f 



= r(sec.^ oj + cosec.2 ic)c?a; 
= tan. X — cot. X. 
dx 



a -\-h cos. X 

dx 



a sm.^ - + COS.-* ^ M~ ^ ^°^- ^ — ^i^- 



2 ' 2/ ' V 2 2 



sec.^-c^oj 



a -\- h -\- {a — h) tan.- - 



by observing that 



sin.2 - -|- cos.^ o ^^ -^-j ^^^' ^ ^^ cos.^ - — sin.' 



X 



and dividing the numerator and denominator of the result by 

/y» 

cos.^ ^. When a ^ h, the last integral may be put under the 
form 



d tan. 



MISCELLANEOUS EXAMPLES. 
X 



335 



2 



2 1 



a — h\a-\-b.^ ^x a — b \a'-^ b 
+ tan/ - ' ' 



a — b 



J 



tan." 



a — b X 
tan. 



\fx-\-b 2 



a — b 



, , tan.-M ri^ -tan. ^V 



/ 



dx 



^a -[- 6 COS. ic/« > 5 Va^ — 6' 
When a <^b, we have 



tan. 



a — b X 

tan. 



\a-\-b *2 






c?a; 



c? tan. 



+ 6 COS. X Z) — a I 5 _L 



a , ,x 

tan.'' - 

b — a 2 



\/b''-^ 



A/b — a tan. - -\- \/b -}- a 

I ^ ; 



by Ex. 18. 



26. 



/ 



dx 



\/b — a tan. - — \/b -\- a 



dx 



a-\-b^ui,x I ^7 . •'^ ^ 

a -}- VJ) sm. - COS. - 



dx 



a [ sin.2 - -]- cos.-^ -\-\-2o sin. - cos. - 



scc.'^ -- </.r 



?M -j- tan.- J -I- 2/; tan. •;^' 



336 INTEGRAL CALCULUS. 



'H+l 




+ tan. - + 



a- \ 2 a^ 

when ay h; but, if a < &, then 



/ 



dx _2l '^(^^''•i + a^ 

a + 6sm ~a\7 ^r~K' W^^^' 
tan. - + 



1 ^ 2^-a 



^ a tan. ^ + ^ - v^^"^ — a^ 

atan.-4-Z)4->v/62— a2 

202. Rationalization and integration of irrational functions. 

Examples of integration by substitution have already been 
given: we now proceed to show under what conditions cer- 
tain irrational differential expressions may, by proper substi- 
tutions, be rendered rational, and integrated by the methods 

previously investigated. 

p 
Let us assume the form x"^{a -\- hx'^)i dx, in which m, n, p, 

and q are entire or fractional, positive or negative. 

1 

-r. . /z^ — a\n 

Put a-\-bx" = z'^; . • . a; i= f — - 

dx=^ ± — — (2 ^ — a) » dz: 



IRRATIONAL FUNCTIONS. 337 

whence 

nh "" 
Now, if — ^ is an integer, the binomial (z'^ — a) " 

is rational in form, and may be expanded by the Binomial 
Formula into a finite number of terms when the exponent 

TtZ I 1 

1 is positive. Each term of the expansion, being 

multiplied by 3-^ + ^~^c?2, will give rise to a series of monomial 
differentials which can be immediately integrated. 
We may also write 

x'^{a-j-bx''ydx—J X ^ 1 {h ^ax-'')^dx; 

and, by comparing this with the first case, we conclude that 
the substitution of 2*^ for 6 -|- ax~^ will reduce 

/ 7?p p 

x'^-^qih^ax-^yidx 

to an expression that is immediately integrable when 

m -\-\ p 

n q 

1 m -{- I p , 
is a positive integer; i.e., wiien -\- -^ is a negative 

integer. 

p 
Hence Jx"'(a-{-hx"yidx may bo rationalized and inte- 
grated when is a positive integer by substituting z"^ for 

a4-hx": and, when -f- ^ is a noL::ativo intei::er, bv substi- 

n q ' ' ' 

tuting z'^ for h -{- ax"". 

It will be shown in a subsequent section (^'iHliat tliointe- 

grals mav also be found when - is a nei::ative inte«;or in 

43 



338 . INTEGRAL CALCULUS. 

771 I 1 Y) 

the first case, and when [- - is a positive integer in 

n <1 

the second. 

Ex. 1. Cx{a -{- bx) . Here w = 1, n=zl, ^ :=z -, and 

w + 1 . ... 

zzz 2, a positive integer. 

n 

2zdz 



Put a -\- bx ^:z z"^ : ,',x=^ — j-—, dx 



b ^ "'*- — 5 

J x{a -\- bxY dx — — j {z"^ — a)z^dz 

2 c 

=z — J{z^ — az*)dz 

2 fz'^ az^\ 2(a -]- bx)i /a -j- bx o> 



\- bx a\ 
T 5J' 



h' V7 5 / h'- 

/x ' dx 
-7- In this example, 

m = 3, n=.2,^=--\ and'''J^=2. 
Put a^ + ^^ = ^^ ' . * . ^ = (2^ — a^)^, and dx = 






/. x^ax c z' 



= {a' + x'-) 



,A x"" - 2a2 



3 

Ex. 3 / — — 5 • la this case, m==2, '?^ = 2, ^z=:— ^ 

771 I - \ ^ . . 

and 1- ^ =: — 1, a negative integer. 

n q 

\a' + x^f . 



IRRATIONAL FUNCTIONS. 

Let 1 -\- a'^ x-"^ z^ z'^ \ ,',x^=^— — 



azdz 



/x'^dx 1 ndz 11 



,2 






^^•^ TT n ^ ?5 1 



Ex. 4. f ^ 1 • Here m = -2, n = 2,l=z 



Put 1 + x~'^ =z z"^: . • . a? = J , c?^ = 



y 



Functions in which the only irrational parts are monomials 
can always be rationalized and integrated. Thus, suppose it 
is required to find 

l>(l-^x^ -x^)dx 

J I 

1 +x^ 

Put x = t^', ,' . dx :=! Gt^dt ; and we have 

Jl _|_ x^' - x^)dx _ w 1 -\.t^- t' Wdt 

J—r^j- -J -i+t^ 



+ 

/ . 1 

4 < 5 

Avhicli becomes the integral in terms oi' x by replacing t by .r\ 



340 INTEGRAL CALCULUS. 

The rule to be observed for rationalizing sncli expressions is 
to substitute for the quantity under the radical sign a new 
variable aJGfected with the least common multiple of the indices 
of the radicals for its exponent. 

Fractions in which the only radicals are the roots of the 
same binomial of the first degree may be reduced to the case 
just treated. 

For example, required 

2 

Cx^ -f {ax -{-hy dx 
X + {ax -f hf 

Assume ^ ^, = ^^.. , . .= '1^ , a.= '^, 

a a 

{ax + hf = t\ {ax + bf — t\ 
By these substitutions, the expression to be integrated be- 
comes the rational fraction 

^PS^{t^ -Vf j^ aH'\dt 
a^ t^^h + at' 

The general method of integrating rational fractions will be 
investigated in the next section. 

EXAMPLES. 
r dx _ . _i 3 + 2a; 



2. j xHxdx 



Ix — 



3. Jd sin. Odd = sin. d — 6 cos. d. 

/dx 
=tan.-ie\ 
gx_[_g-a; 



/3 
(1 — cos. xydx= ~x — 2 sin. x -f- 



sin. 2x 



6. 



MISCELLANEOUS EXAMPLES. 341 

x'^dx 1 a^ -\- x^ 



/x^ax i , « -r 

a^ — x'^~Qa^ a^ — 

. rl -{- COS. a; 7. , . N 

7. / — ^^— ^ ax:=l(x-{-8iii.x). 

J X -\- sin. a? ^ ' ^ 

/•rc + sin. a; aj 

8. / ^i — j ax z=z X tan. - • 

^ 1 + COS. a? 2 

9. f-^ c^a; = Ix lilx) - Ix. 
•^ X 

10. / e^'^sin. ma:;cos.?ia:c?a; 



e"^ a pin.(m + n)x — {m -\- n)cos.(m -f- n)x 
+ 



e"*-^ a sin. (m — 7z)x — (m — n) cos. (m — 7z)a; 



2 a'' + (m — 7i)2 

Having found the indefinite integral, the definite integral 
between assigned limits, except in special cases, can be at 
once determined. 






Tta' 

T 



for / ^/a' — a^'^cZa; = ^ + -r sm. - = v<J:), 

*^ A A a 

and v'(«) = -r' M0) = 0: . •. T/,(a) — v(0) = ^. 



Jn2rt _^ 

ver.~^ - cZa; = na. 

By making x =^ a{\ — cos. z'^), wo find 

/vor.~^ - dx = I aO sin. OdO 
a »^ 

:=. a sin. f^ — aO cos.tK 

The limits tt. and (or tlio transformed intog-ral correspond 
to the limits 2a and for the given integral. 



342 



INTEGRAL CALCULUS. 



13. 
14. 

15. 
16. 

17. 



/ X ver. ^ - ax 
•^0 a 



sin. OJ -}- COS. ic y^2 
sin.2 icc^ic 



5;ra^ 

1 _ (x n 
= _aan. - + 



/ 



a -[-Z)cos.^a? 



N ah' s/a 



Va tan. ic _ ^ 



/. V^H^^cZ. = ("4;?^ - ^^ {a + 5.^)1 



2a + a^ 



riia-j-a: 
./ a -4- a; 



a — X 



-h^ Na + ic 



cZa? z= V^ 



a-— X' 



2a>/a — 5 
V« + X 



18. / {a-^hx^'Yclx 4 

t/ ■ ziz -^ — (d 

X 371^ 



+ bx"' 



In effecting this integration, transform by assuming 



SECTION II. 

INTEGRATION OF RATIONAL FRACTIONS BY DECOMPOSITION 
INTO PARTIAL FRACTIONS. 

203, A RATIONAL fraction is of the form 

A' -^rB'x-\- G'x^^ + Wx''' 

in which the numerator and denominator are entire and alge- 
braic functions of x ; the co-efficients A^ B ..., A\ B'..., being 
constants. 

Denote the numerator of such a fraction by F{^x), and the 
denominator by f{x). If the degree, with respect to x, of 
F(^x)^ is not less than that o^fi^x), we may divide F{x) by /(a;) 
until we arrive at a remainder of a degree inferior to that of 
f{x). Let <:p{x) be this remainder, and Q the quotient; then 

'F(x)dx -^^, rq (x)(Lv 

and ■ — / /I./.. I 



,1 r^4?)«^=rcw.+ r^ 



(X) 

As jQdx can always bo found, the integration of tlio oiigi- 

(r ( x^iix 
nal fraction is reduced to the inte<;*ratI(,Hi iif — , in whli^li 

the degree of cf{x) is lower than that o[\/\.v). Tlio integra- 
tion of - is eflVctod bv rosolvlnir it into a series o{ more 

/(■'■) 

simple fractions, eahed j[)rnV/'()f? yVcuY/o/^v ; and wo will now 
demonstrate the possibility oi^ sucli resolution in all eases in 



344 INTEGRAL CALCULUS. 

wliicliy(ic) can be separated into its factors of tlie first degree 
in respect to x. 

F{x) 
204, Suppose tiie fraction — — - to be in its lowest terms, 

and tliat the degree of F{x) is inferior to that of fix). If the 
factor X — a enters /(a:) jp times, we shall have 

f{x) = {x- ay<fix), 

q){x) denoting the product of the other factors of/{x) : whence 

F(x)^ Fix) J^""^ ,{a)''^\ cpja) 
/(x) {x — a)^(f{x) {x — a)Pcp{x) ~^ (x — a)^' 

But F(x) — q](x) = when x=za, and is therefore divisi- 

(p(a) 

ble hy x — a. Let li^i(ic) be the quotient, then 

F{x) _ ^,{x) .F{a) 1__ 

/(x) ~{x — a)P-^cp(x)~^ cf.{a) (x — ay' 

Denoting by A^, we have 

cf{a) 

F{x) _ ^iW ^1 . 

/\x) {x — a)P~^ ^{x) '^ {x — a)P ' 

that is, has been resolved into two parts, one of which is 

f{x) ^ 

'—^. In like manner, '.^^ 

(x — a)^ ' (x — a)P ''(•f{x) 



-. In like manner, l^r^^ — 7— r niay be reduced to 



yi)x{'x) '^2(^) i_ -^ 



+ 



(ic — a)^-ifp(ic) {x — a)P-''^^{x) ' [x — a)^-^' 
and so on, until at last we should have 



[x — a)q)(x) cp[x) x — a 



Ap 



RATIONAL FRACTIONS. 345 

By the successive substitution of these values of 

; ^\^ \ — --, , ^l\ { — 7-^..., in the order the inverse 

of that in which they were deduced, we find 

F{x) ^^Pp{x) A, A, Aj, 

f{x) ~cp(x) '^ {x — ay^ {x — ay-'^~^ ^x — a 



lb ( X) 

Proceeding in the same way with the fraction ' ^ , the 

F(x) 
decomposition of - — - into partial fractions will be at last 

completely effected. 



205, If the root a is imaginary, and equal to a + 13>/ — 1, 
then «!=:« — /3\/— 1 is also a root of the equation /{x) = 0. 
Suppose that all the partial fractions corresponding to the real 
roots have been determined, and that there remains for resolu- 

tion into such fractions the fraction —— ^ — , in which y^x)=rO 

gives rise to imaginary roots alone. Suppose, further, that the 
pair of roots, x =^ a -{- §\^ — 1, a; = a— /5V' — 1, enters this 
equation (2' times. Denote the factor of /i (a:) that gives the 
remaining imaginary roots by cpi{x) ; and, to abridge, make 
a = a -t- pV — 1, «! = a — pV — T: tlicn 



= a: - a_ cr,^,a) 

{x — ayi-\.i: — aiyi ^ (x - a)''(J- — «,)« ^ ' 

44 



346 INTEGRAL CALCUL US. 



In like manner, 






X — a 



(x — ay-'^{x — a)icf^{x) 

iiM _ ^i(^) cpjx) 

X — a (^1 — ^)<jPi(<^i) 

■"" ~ {x — ay-'^ {x — a^y cpi{x) 

qpi(«) 

■^ {x — ay-\x — a^y ^ ^' 
TEe last term in the second member of Eq. 2 may be written 

g^i(«i) <jPi( Q^) 

(ai — a){x — ay~\x — ay' 

But (ai - a) = - 2V^^ and :?i^ is derived from '^^^ 

<Pi{«i) g'i(«) 

by changing the signof V— 1: hence, if -—-{^ — ^ -|- B\^ — 1, 
then ^i^^ = ^ - ^v/=a; and 

g)i(ai) (jPi(a) 
Therefore 



(jPi(fti) g)i(a) 



(ai — a){x — a)«-^(cc — ai)-? (x — a)^^-\a; — aj)'^' 
and the sum of the two partial fractions in the second mem- 
bers of Eqs. 1 and 2 will become 



RATIONAL FRACTIONS. 347 



{x — ay{x — a^y ~^ {x — ay-'^ {x — a^y 



A + BV-l + -\(x — a)-^V-l\ 

^ r ) 

[x — ay^{x — a^y^ 



~ {x''-2ax-{-a''^[i''y 

which is rational. It is also seen, that the numerator and de- 
nominator of the first term in the second member of Eq. 2 is 
divisible hy x — a. Dividing, and denoting the numerator of 
the result by ^i{x), this term may be put under the form 

X (x) 
^^ : hence, by substitution in Eq. 1, 



{x'^ — :^ax + a^ + ^-y-^ cf^{x) 

we should have ^ 

T7 / N n/ / N ~-(x— a) 4- A 

Fx{x) ^ ^i(^) _r [___ . 

f^{x) {x'' — 2ax-\-a'-\-^''y-^q^^{x)^{x'-'lax + a'-\-f-y 

Now, Zi(x) is a rational and entire function of x, and the frac- 

X (x) 
tion 7—, -r — ^--r-i — PTTTr-j t^ mav be treated as was 

{x^ — 'lax -\- a^ -\- f-y-^ q ^[x) 

jp / \ 

. ^ , and so on : our result with respect to the assumed pair 

of imaginary roots being of the form 

F,{x)^X^^(x) 3f,x + N^ 

J\{x) cp,{x) "^ (x'' - 2cu:. + a- + p^j'^ ^ 

M,x-^N, . 



' X-— 'lax-^a^ -f p^ 
The possibility is thus demonstrated of resolving a fraction, 
the terms of which are rational functions of a single variable, 
into a scries of rational partial fractions wlicnevcr the denom- 
inator of the given fraction can be separated into factors, 



348 INTEGRAL CALCULUS. 

whetlier real or imaginary, of the first degree with respect to 
the variable. The investigation also shows the form of the 
partial fractions answering to the different kinds of factors of 
the denominator of the given fraction. 

F(x) 
Thus if fix), the denominator of - ^; / , contains the factors 

f{x) 



X 



— a, {x — b)"", {x — cy, {x — a — ^V~l)P, 



(a; — a + pV— 1)^ x — y — d\/—l, x — y + d\/—l^ 
then 



/{x) x — a (x — by {x — by-^ ^ x — b 



{x — c)" [X — Cf~^ ' X — c 

M^x-\-N^ M^x-^N^ 

~^ (a^2 _ 2ax + a^ + f)P + {x'' — 2ax + a^ + 1^2)^-^ 

"1 f" ^'i _ 2aaj + a^ + /^^ i" ^2 _ 2^,5 ^ ^2 _^ ^2* 

The labor of determining the constants Ai, ^2-"-^i----^i; 
j^^..., for the partial fractions, which, by following the method 
above indicated, would be very great in many cases, may be 
diminished by expedients which we will now investigate. 
The most obvious of these is based on the consideration, that, 
when the partial fractions are reduced to a common denomina- 
tor, the numerator of the result is identically equal to the nu- 
merator of the given fraction. 

20S» To determine tlie partial fraction corresijonding to the 
single real factor, x — a, off(x). 

Assume -^ = h -^ (1), 

fix) x-a^<^ix) "• ^' 



RATIONAL FRACTIONS. 349 

in which ^ is a constant, and —7^/ is the sum of the partial 
fractions answering to the remaining factors of f{x)j and 

From (1) we have 

F{x)=Aci{x) + (^x-a)xp[x) (2), 
an identical equation. Make a; =r a^ and then 

Fia)^Acp(a): .-. A = -^^, 
We also have the identical equation 

/(^) = (^ - ayp{x) ; 
whence^ by differentiation, 

f^x)=.cf(x) + {x-a)q>^(x), 
an equation also identical : therefore, making x = a, 

207. To determine the j)artial fractions corresponding to the 
real factor (x — a) repeated n times. 
We now assume 

/{x) {x-a)''^{x — ay—^'^ x-a~*'cfix) ^ '' 

— }— .- dcnotinc: the sum of the fractions to which the other 
<ip{x) ^ 

factors of f(x) give rise. 

Multiply botli members of (1) by (.r — (0", and wo have the 

identical equation, 

^~^A,^A,{x-a)Jr"'-}-^i„i^^r-a)- ^ -^^'^''\{:v - aY, 
observing thaty\.r) r- 9 (.r)(^.r — if}". Denoting the first nicni- 



350 INTEGRAL CALCULUS. 

ber of this equation by x{x), and then, in it and its successive 
differential equations, making x^^ a^ we bave 

X{a)=A„ x'{a) = A„y/'{a) = l.2A,..., 
;£(»-'>(a) = 1.2...(n-l)J„, 
and thus the numerators of the partial fractions are determined. 
208, To find the partial fraction corresijonding to a single 
pair of imaginary factors. 

Suppose X — a — ^V— 1; x — a^ ^V — 1, to be the ima- 
ginary factors oi f{x). We then put 

F{x) _ Mx-\-N w(x)^ 

f{x) ~ x^ — 'lax -|- a^ + pi'^ Qp{x)' 
whence F{x) — cp {x) {Mx + N) + xp (x) {x"" - lax + a"- ^P), 
an identical equation. 

Make a; = a + /3 >/ — 1 ; then 

or, by making ic = a — (5V — 1, 

F{a - pV^^n:) =r g)(a - |3 V^=l) { M{^a - /3\/^n^) + N\ - 
These last equations may be written 

A + Bs/^^ ( C 4- Bs/'^^) \ if(a + pV^^l) + N\, 

J _ i^ V^^ = ( (7 - Z>\/:rT) { i¥(a - /5 v/=l) + 7\^j , 
in which A^ Bj (7, and D are known functions of a and §, 
From either of these equations, the values of 31 and N may 
be found by equating the real part of one member with the 
real part of the other, and the imaginary part of one member 
with the imaginary part of the other. 

The values of M and N may also be found by the method 

of Art. 206. Thus, for brevity, denote the imaginary factors 

by a; — a, x — ai ; then the partial fractions are 

F{a) 1 F{a,) __1__, . ' . 

f {a) X — a f'{a^) x — a^ 



RATIONAL FRACTIONS. 351 

If JJ^^ - ^ + W^, then 5|^J = A- Bs^^l, since 

\ 4s derived from by changing the sign of V — T : 

hence, replacing a and a^ by their values 

a + ^V^ a-fV=3; 
the fractions become 

ic — a — |SV— 1 x — o:-[-pV^^' 
the sum of which is 

2A {x — g) + 2B^ 
x^ — 2ax -[- cx^ 4~ A'^ 
j^09. To find tlie partial fractions corresponding to a pair 
of imaginary factors which enters the denominator of the given 
fraction several times. 

Let X — a — §V— 1, X— a-{- ^V— 1, be the imaginary fac- 
tors, and, to abridge, put a=: a-\- pV —1, a^ = a — pV — 1 ; 
then, putting/(a;) — \{x — a){x — aj) j'^r/:(x), 

f(x) ~" i {X -a){x - a,) j '^ "^ I (X -a) {x - a,) j '^"^ "^ ' " 

M^x + N,jX xp{x) 

\{x — a){x — ai)\ cfix)' 

representing by -|\| the sum of the partial fractions to 

which the remaining factors ofy^.r) give rise. ^lultiply the 
first member of this equation by /\.r), and tlio second member 
by its equal j [x — a)(x — a^)\'^ (jr(.r), and wo have 

F{x) = {:}f,x + N,)ciP') + [^%^^' + ^^^H'^• - ^r\^v - a,\r^x) 
+ {M,x + K.) j (.r — a){A' - a,)\\yi^x) H 



352 INTEGRAL CALCULUS. 

Now, whether we make a; = a, or x^^a^, all the terms in 
the second member of this equation, except the first term, 
vanish. Suppose ic = a, then 

and if the real parts in the two members of this last be 
equated, and also the imaginary parts, we shall have two 
equations from which to find the values of 31^ and N^. Sub- 
stitute these values in (1), transpose [MiX -\- N{)q}(x)j and 
divide through by {x — a){x — a^), denoting the first member 
of the resulting equation by F^(^x) ; then 
F^{x) = {M,x-^N,;)(f{x) + {31^x-^N;){x — a){x — a^)^{x)-\-''' 
^\^x-a){^x-a{)\^-\{x) (2). 

Proceeding with (2) as we did with (1), the values of M^ and 
^2 niay be found ; and, by repeating these operations, all of the 
constants, M^, Ny, M^, N^^..., will finally be determined. 

210. The rational fraction, which may be decomposed into 
partial fractions by the foregoing methods, being a diff'erential 
co-efficient, the resulting fractions are also differential co-effi- 
cients ; and the sum of their integrals will be the integral of 
the given fraction. 

The differentials corresponding to these partial fractions are 

of the form 

Adx {Mx -\- N)dx 

(x — a)"'' {x'^ -\- px -\- q)" 

1 A 

The integral of the former is -^ -— ^i;— ->, which 

^ m — 1 {x — af-^^ 

becomes Mix — a) when m = 1 ; and that of the latter, when 
ri = 1, has been explained in Ex. 20, Art. 201. The integra- 
tion of the second form, if n be greater than unity, is reserved 
for the next section. 



RATIONAL FRACTIONS, 



EXAMPLES, 



1. ^-r, ^ ^= -J^' The factors of the denominator are 

x^ — X — "Z f{x) 

X -\-l, X — 2. We therefore put 

3 — 2a? _ A Ji 

x'^— X — 2 cc + li^ — ^ 

Substituting — 1 and + 2 successively for x In 
P(^x) __S-2x 

we get 3^ ^^""3* 

(3 — 2x)dx _ 5 dx _ 1 dx 
x^ — x — 2~ ~ I ^"+I~3it;— 2' 

^ ^ (a;2 — 3x — 2)dx 
"' c/ (x"^+ic + l)-(ar4-T)' 

In the denominator of this, tlie pair of imaginary factors, 

x-\-~ — - V— 3, ^ + TT + ,) V — 3, enters twice ; and the real 

A A A A 

factor X -{-1 once. We put 

x"" - Zx - 2 __ J^ix^ N, 3Lx + .Y, xiix) ^ 

{^^'~or-\-~\f{x+\) ~ {^x''j^x-\- If "^ ^M^"+ 1 ^H^l * 

x''-^X-2 =:(J/i.l' + Xi)(.l' + l) 

+ (il/^o; + N,)(,x'' + a; 4- l)(.u + 1) + l-^'' + ^^' + nV"(a') [^W 

Give to re one of the values wlu'cli reduce x'- -\- x -\-\ to zero ; 
then, for this vahie, (1) becomes 

a;^ _ Zx - 2 = (.v.,,- + X,\.v. + 1 ) (ilX 

45 



354 INTEGRAL CALCULUS. 

From x^- ^ X -\-l = , ^vQ have x"^ =^—x — \. Substituting 
in (2), 
— 4.x—Z=, M^x"- + M^x + N^x + N^ 

^M^{~x-l)-{-M^x + N^x-^N^ 
= -M^-^N^x + N^'. 
whence M^ _ iv; = 3^ J\r^ ::= _ 4, 3£^ =: - 1. 

In (1); replacing 31^ and iVJ by the values thus found, and 
transposing, we have 

ic2 — 3a; — 2 + (x + 4)(^ + 1) == 2(^2 + 0^+1) 
=: {M^x + iVOC^' + ^ + 1)(^'^ + 1) + (a;2 + a; + l)2a/;(a;) (3). 

Dividing through bj ic^ -|- a; + 1, and in the result making 
a:^ + cc 4- 1 == 0, we get 

whence J/2 = — 2, JVj = 0. 

\ij (x\ 
The partial fraction corresponding to --^-V-4 may be found 

X — 1~ X 

by the method explained in Art. 206 ; or thus : After dividing 
(3)bya:;^ + a; + l, replace M^ and N^ by their values, trans- 
pose, and again divide by x'^ -\- x -\- 1. We find ip{x) =z2\ 



therefore 



{x'^ — Zx—2)dx r (^ + 4)d^• 



r [x^ — 6x — :^)ax _ r 
J (x^'-Ux-hlfixA-l) ~~ ~J 



(^x'^j^x-{-lf{x-\-l) J {x'-\-x+l)'' 

2xdx . r 2dx 



/zxdx r 

x'4-x4-l'^Jc 



aj^ + aj + l J x-\-l 
g r(^x^_+dx — 12S)dx 

J x^ — 5x^ + 3x-^d ' 
By the method of equal roots, we readily discover that the 
denominator may be resolved into the factors (x — 3^, x -{-1: 
hence we put 

9:c2 + 9a;-128 A. , A , ^i 



i^3 _ 5^2 _|_ 3^ _|_ 9 (ic — 3)2 ' x — S ' x^l' 
whence 



RATIONAL FRACTIONS. 355 

from which, by making oj zz: 3 and cc = — 1 successively^ we 
get Ai= — 5, ^1 = — 8. If the second member were de- 
veloped, the co-efficient of x"^ would be ^2 + ^i 5 equating 
this with the co-efficient of x"^ in the first member, we have 
^2 + ^1 = 9: ,*, A^ — ll ', and therefore 

r {9x''-{-dx- 12S)dx _ _ r 5dx r lldx r Sdx 

J ic=^ — 5:^2-1-3x4-9 "~ ~J {x — 3)2 '^J ^^^ ~J S"+l 

-f 17Z(a; — 3) ~Sl{x + l). 



x-3 



/^ »i — 1 ^^ 

211, Integration of — j- when m and n are positive 

X — X 

integers. 

If n be an even number, the real roots of a;'' — 1 = are 

-|- 1 and — 1 ; and the imaginary roots (Art. 77) are given by 

,, . 2^77 , / — -. . 2kTt , . . , 7 . 

the expression cos. =b v — 1 sm. , by giving to fc in 

n n 

succession the values 1, 2, 3..., 1. 

A 

Tt Fix) 

We will denote the arc - by d, -— ^ beino- the fraction to 

to be resolved into partial fractions. It has been shown 
(Art. 206), that, if a be a root of /(x) = 0, the corresponding 



F(d) 1 X 

partial fraction is -—-r : hence, for the fraction — 



m — \ 



f(a) x — a ' .r"-l' 

the partial fraction for the root -}- 1 of llio 0(|uation .r" — 1 ir= 
. a""-^ _ a"' __ 1 
na " ~ ^ na" n{x — 1 ) 

(—1)"* 
fraction is ■, — ,— rr- The i)air of iinairinarv roots 
n{x-\-l) ^ 

cos. 2/j'/rt V-ZT, sin. 2/.-^^, 
give tho partial fractious 



IS .- = — -=: — -— ; and, for tho root — 1, the partii: 



356 INTEGRAL CALCULUS. 

( COS. 2ld + V^ sin. 2hd) '^ 
n{x — COS. Ihd — V— Isin. 2hd) 

, (cos.2A;^ — V^^ sin. 2^^)"* 
that is, (Art. 73), n(x-cos.2fc. + v/=Tsm.2fc.)' 

COS. 2rf^ + V^^ sin. 2m^'^ 



n(a; — COS. 2kd — \^ — l sin. 2A;^) 

COS. 2mkd — V— 1 sin. 2rf^ 



?2(a; — COS. 21cd + V — 1 sin. 2kd) 

_ 2 COS. 2mhd(x ~ cos. 2^^) — 2 sin. 2mhd sin. 2^^ ^ 

~ n(x^ — 2iccos. 2A-^ + 1) ' 

and for each pair of imaginary roots, that is, for each of the 

77/ 

values 1, 2, 3..., - — 1 of A:, there will be a partial fraction of 

the form of this. Let the symbol 2 denote the sum of these ; 
then 

""-^dx __ r dx H—iy^dx 



/x'^-^dx __ r dx r 

ic"— 1 ~J nix - T) "^ J 



{X -I) 'J n{x+l) 

^11 



2 ^ COS. 2mhd{x — cos. 2hd) — sin. 2m'kd sin. 2hd 
{x - COS. 2hdf -f sin.2 2kd " 

:^Llloo-l)-\-^—^^l{x^l) 

n ^ n 

-f - ^ COS. 2mM(x- — 2x cos. 2^^ + 1) 
n 

'^ ^ • 7«+ -i^^ — COS. 2^^ 

^ sm. 2mA;^ tan. ^ , — ^r^— — ? 

n sm. 2A;^ 

by observing that the last term under the sign of integration 

can be separated into the two fractions 

2 COS. 2m'kd{x ~ COS. 2kd) 2 sin. 2m^^ sin. 2^^^ 

n'^ x'' — 2x cos72k9 -{- I ~n {x — cos.2kdf-\-sm.''2kQ 



X' 



dx 



212. Integration of — r— , m and n being positive inte- 

X ~~" X 

gers, and n an odd number. 



EXAMPLES. 357 

In this case^ x^ — 1 = has but one real root, -|~ 1 5 ^^^^ ^^^ 

imagmary roots are the values assumed by the expression 

^hit , / — - . ITcTt ...,,. • .-. 

COS. - — rh V — 1 sm. -, by giving to h m succession the 

n — \ 
values 1, 2, 3..., — ^ — ■ (Art. 77). Hence, by operating as in 

the preceding article, we find 
•^__^^^ = l(^jc-\)-{- "i 2 cos. 2mhdl{x' - 2x cos. 2kd + 1) 

2^- o7«^ _i^ — cos. 2Jcd 

— ^ mil. Zm/cd tan. ^ ; • 

n sin. 2Jcd 

x"^ — ^ clx 
213, Integration of ^ ; m and n being entire and 

i^ — |— i 

positive, and n even. 

Under the supposition, none of the roots of i:c" + 1 == are 

real ; and the imaginary roots are found by giving to h, in the 

2^ + 1 , . .2/^ + 1 ^, , 

expression cos. — tt =b V — 1 sin. — - — n, the values 

0, 1, 2. . . ^ — 1 in succession. Put ^ for - , then the partial 
2 n ^ 

fractions corresponding to a pair of these roots will be 

cos. {21 + 1)^ + \/^^=Tsin.(27g -f 1)^ 
X - cos.(2A; + 1)^ — \^'^^i^m.(2h + 1)^ 

, cos.(27g + 1)^ — \/^=n[sin.(2/g + l) d 

X — cos.(2/j + 1)^^ + V^— rsin.(2/^ -^V^i 
the sum of which is 

^ 2 COS. m{2h-\- \)d j x — cos. (2A:+ l)o\— sin. m{2k -f 1)^ sin. (2i--[- 1)9 
n J X — COS. (2^- -[- 1 )</ j ' -|- slu.'^ (2A- -j- 1)^ 

Hence 

/QQ^n — 1 ^j, 1^ 

^^Tqrr = - n -''''''- '"<-^' + ^ )^^^ i '"'" - -'^•^^os.(^2A' + 1 V^ -h 1 i 

-4- Jl sin.mi 2A' -4- I VMaii. ^ - ^ -— . 

^ n ^ ^ sin.^2A--f 1)0 



358 INTEGRAL CALCULUS. 

In like manner, we integrate — ^ , when n is odd, by 

OC —\ — X 

finding the partial fractions corresponding to the roots of 
a;" -|- 1 =r 0. In this case, there is one real root, — 1 ; and the 
other roots, which are imaginary, are the values assumed by 
the expression 



cos. (2h + 1) - ± V- 1 sin. njc + 1) -, 

^ 1 

by giving to h the values 1.2.3... — - — successively. 



"We should find 



— -^cos.m(2;^+l)^jic2_2xcos.(2;t +1)^+1 

, 2^ . ,o7 . 1^^+ _ia7^cos.(2Aj + l)^. 

~\ — ^sm.m(2Aj + l)/9tan. ^ ; — —^ ^^ — — 

~n V -r ; sm.(2A; + l)^ 



EXAMPLES. 



dx _1 Jx-Vf 1 _i2a; + l 

— 1-6 ^^qi^+l~V3 

dx 1 -1^, 1 I a ^ X 



r dx _i J {X— iy . 



J a^ — a?^ 2iJ^ a 4a^ a — cc 

^- / ^ 2 o = -^^ ^ — rr + Q V2 tan. 1 -— 

^ r x'^dx _1 /9 7^^ — xa,/2-}-1 

J^M^~4^ ^^ + 0^^2 + 1* 

+ ^ V3 { tan.-X2aj _ ^3) _ tan.-i(2x + ^3) j . 



EXAMPLES. 

/dec 
i' This may be rationalized by putting 
{i — x'Y 

1 — X^ =:X^Z^] 

whence dx= j, (^l — x^)^ = ^ .* 

{z' + if (i + «T 

dx r zdz 



r dx __ r zdz 



in which, if we replace z by its value — -? we have the 

X 

required integral in terms of x. 



sectio:n^ hi. 

FORMULA FOR THE INTEGRATION OF BINO]knAL DIFFERENTIALS 
BY SUCCESSIVE REDUCTION. 

214, The integration of differentials of the form 

may be made to depend on that of other expressions of the 
same form, in which the exponent of the variable without the 
parenthesis, or the exponent of the parenthesis itself, is less 
than in the original expression. This is accomplished by the 
method of integration by parts. T7e have 

Jx^'i^a + Ix^ydx =j'x'"-«+i(a + Lx'^yx'^-'^dx =fudv 
n, -1-1 (a + 5^")^+i 

and therefore 

The integration of x^'^a -|- hx'')^ is thus brought to that of 
ic'"~''(a + 5x")^+\ which last is more simple than the first 
when m is positive, and greater than n, and whenj9 is negative; 
for then the numerical value of 2^ -{-'^ i^ l^ss than_p. 

But we may find a formula in which the exponent of the 
variable without the parenthesis is diminished, while that of 
the parenthesis itself is unchanged. 

Thus we have the identical equation 

ic'^-"(a + 5x")^+i 1= x'^-^i^a + hx'')P{a + Z?a^") 

= aa;"*-"(a + hx'')^ + hx'^{a -f hx")^', 



REDUCTION FORMULAE. 361 

therefore 

+ bjx"'{a + hx'')Pdx. 
Substituting this value in Eq. 1, we have 
fx^ia + hx-ydx == a;— " + i {a + hxy + '' 

- ^tP^I ^ fx^{a + hx-Ydx; 
whence, by transposition and reduction, 

J ^ ^ ' b{m -{- ri'p -\- \) 

aim — n-\-V) r 

— rr-^\ ^^Jx^~\a^hx^^Ydx (A). 

b{iin -[- ^?i> + 1 ) '^ ^ ' 

The integration of x^\a -\- bx^ydx is then made to depend 
on that of a;"'~"(a + bx'^ydx; and, by another application of 
the formula, the integration of this last reduces to that of 
a;"'~2''(a 4- /;j?")^'(ix, and so on: hence, if ^m is positive, and 

greater than n, and i denote the entire part of the quotient . 

the integral to be determined after a number, /, of reductions, 
would bo 

J x'''~"\a -\- bx'')^' dx. 

If m — in= n— 1, this expression is imnuHliatoly intogra- 
ble ; f )r 

but m — in = n. — 1 leads to l= / 4- 1 , and iho condition 

of integrability (Art. llOl!) is then sadsfuHl. 

46 



362 INTEGRAL CALCULUS. 

Formula A cannot be applied when m -j- ??/? ~[- 1 = ; for 
tlien its second member takes the form oo — oo : but in this 

7Yl -\- 1 . 

case 1- j9 is equal to zero, that is, an entire number; and 

the original expression is therefore immediately integrable. 

215* Formula for the reduction of the exponent of the 
parenthesis. 

Assume 

x'^ia-^'bx''ydxz=i(a-^hx''yd ^^^—^ = udv ; - 
then, integrating by parts, 



Jx'"{a-{-hx'')Pdx= - 



m + l 



(a -h hx^y 



m -\- 1 
- -^^ fx'^+^a + bx^^y-'^dx (1). 

In this formula, the exponent of the binomial has been 
diminished by 1, while that of x without the parenthesis has 
been increased n units. We may, however, diminish the for- 
mer without increasing the latter exponent. In formula A, 
last article, change m into m + ^^; andp into^ — 1 : we thus 
have 

^x [a^bx ) ax- j^.^^_^^^_^ j,^ 

b{np + m 4- 1 ) •^ ^ ^ 

and this value of J'cc'^+" (a + Ix'^y-^ dx^ substituted in Eq. 1, 
gives, after reduction, 

fx^ia + hx-ydx = ^":^'(^ + ^^")'' 

•^ ^ np -\- m-{- I 

712^ -{- m ~\- 1 ^ ^ ' ^ 



REDUCTION FOUMULJE. 3G3 

By the repeated application of this formula, the exponent p 
will be diminished by all the units it contains. This formula 
will not admit of application when ^ip -|- m -f- 1 = ; but then 

the integral fx"^{a + hx^ydx can be found at once (Art. 202). 
By means of formulae A and B, the integral fx'^^a -\- hx^ydx, 
when m and p are positive, may be made to depend on the 
more simple integral fx^'^~'^'^(a -\-bx^)^~'^dx ; in being the 
greatest multiple of n less than m, and q the entire part of p. 

216. Formula for the reduction of the exponent m Avhen 
m is negative. 

From Eq. A, Art. 214, by transposition and division, we find 

aim — n-\-l) ^ ^ ' 

Changing m — n into — m, this becomes 



Jx-''\a-\-hx'')Pdx — 



m + l 



{a-^hx''y + 



a{m — 1) 

a[m — 1) '^ ^ ' ^ ^ 

If in denote the greatest multiple of n contained iu m^ then, 
by i -|- 1 applications of this formula, the integration of 

will depend on that of .r ^ "' + ^' + ^'"((i -|- /j.r"V'(/.r ; ;iml. if we 
have — m -\- [l -^ \)n — n — \ ^ 

we have /-.r" \a + />.r"V'.^.. -- ^'' + ^^'^■"^""'\ 

7^ , 1 .1 • •,• • — /^/ 4- 1 

i>iit under this sni)iH)sit ion, siuci> = — / an tMitire 

n 

number, the original expression is iiuuiediately integrable 
(Art. 20:^). 



364 INTEGRAL CALCULUS. 

217, Formula for the reduction of tlie exponent p when p 
is negative. 

From Formula B (Art. 215), we find 

fx'^ia + Ix'^y-^dx — — — ^— ^- '— 

^^ ^ ^ anp 

' anf »/ ^ ' ' ' 

and, if in this we replace p — 1 by — i^; it becomes 

/x"Ya + Ix'^yPclx = — - .~ -A 

•^ ^ ^ ani^p — 1) 

By the continued application of this formula, the exponent 
of the binomial will finally be reduced to a positive proper 
fraction. When ^9 = 1, it cannot be applied ; but then the 
integration of the given expression may be brought to that of 
a rational fraction. 

218, The preceding formulce facilitate the integration of 
binomial differentials j but it is to be observed that the exam- 
ples to which they are applicable belong to cases of integra- 
bility before established (Art. 202), and the results may there- 
fore be obtained independently. 

By the application of Formula A, we have 

. r x'^dx _ x^'-^s/l — x'^ m — 1 rx'^-'^dx, 
J VT— x'^ ~ m ' m J \/l — x'^^ 

and, by making m == 1.3. 5. . . successively, this gives 

/xdx , 



X" 



r x^dx _ x"- 2 r xdx 

j TT^^ - - 7 v/l - c.' + 3 j ^^==p, 



REDUCTION FORMULAE. 365 

/x^dx x'^ ^ 4 r x^dx 
vr^^ ~ " "5 '^ + 5 •' vT=^ ' 

whence 

/• x^dx /a;* , 4a;^ , 2.4 \ , 

^ Vr=r^== ~ (,^ + 375 + 1X5;^! - *'• 

When m is an odd number, we have the general formula 

/x'^dx 



( m "^ (m — 2)m "^ "^ 1.3. ..m \ ^^' 

/: 



and, when m is an even number, 
x'^dx 






X--' {m-l)x—^ 1.3.5...(m-l) J .^ -, 

m "^ (m-2)m "^ ^ 2.4. 6.. .71 ^ 



, 1.3.5...(m-l) . , 
2.4.6...77i 



dx 



2. f "^ , = fx-"'{a' + xT^dx. 

^x^^ia^ + x'^y ^ 

Comparing this with Formula C, ami making 

we find 

i 



r dx __ _ (a^ + x"") 



m — 2 /» dx 



366 INTEGRAL CALCULUS. 

Without referring to the formula, this expression may be found 
as follows : — 

(a' + x'f n a'-\-x' 

= ^-— "- — - + (m + 1) / -L ^ dx; 

^m+i ^ x'^+\a'' -\- x'f 

whence, by transposition and reduction, 



/> dx (a2 + a:2)2 r dx 
(m+ l)a^ j Y= ' mj 

n dx _ _ (a2_|_^2)^ 



m — 2 C dx 

{m— l)a'' x^'-^a'^ -]- x'^f 

(Mx -\- N^dx 
219, By means of Formula D, the expression / .^ ~ — /—^y 

{x -^px-\-q) 

which occurs in Art. 210, may be integrated by successive 

reduction. 

Let a -\- ^\^— If a — ^\^— 1; be the roots of the equation 

x"^ -\-px -\- q^O; then 

( Mx + N)dx _ (Mx + ]Sr)dx 



\{x-ay + ^'l'' \{x-ay-^li'^ 

Putting ic — a =: 2, we find 
rMix-a)dx ^ 

__ M 1 

~" 2(7^ — 1) (s2 4-/5'2)«-i 

M 1 



2(71 — 1) j(^_a)2_|-^2| 



«-i 



REDUCTION FORMULjE. 367 

Making x — a:=^y, and Ma -{- N= M', we have 
f {Ma+N)dx __ r M' dy _ r _ 

and therefore, by combining these results, we have 
/* {Mx-\-N)dx _ _ M _; 1 

Formula D may now be applied to the term under the sign \ 

in the second member of this last equation ; and, by repeated 
applications, the exponent — n will be reduced to — 1, when 
the integral will be completely determined. 

220, Reduction formulae may also be constructed to facili- 
tate the integration of trigonometrical functions. Let the 
integral of sin. ^ a? cos. ^icc^x be required. 
Make sin. a; zz: z; then 

cos.cc = (1 — 2^)S dx — {\ — z^) ~dzj 

1-1 
and sin.^a;cos.^a;c?x= 2^(1 — 2'^) - dz. 

Now, if q be an odd number, whether positive or negative, 

we may always effect the integration of 2^(1 — 2-) -^ dz, what- 
ever may be the value of jo. In like manner, by making 
cos.x := 2, wo see that the integration can bo effected when j:; 
is an odd number, whether positive or negative, whatever bo 
the value of q. 

In any case in which p y 0, and ^ ., - > 0, by applying 
FormulaQ A and B, we get 

^pn _ 2-2) , ,1, ^ _ ^1_ -J 






p-\-qJ ^ ^ 



368 INTEGRAL CALCULUS. 

g-i 

^^(1 - z"-) 2 c?2 = li Li 

If jp < 0, Formula C gives 

2-:p(1 _ 22)-2-cZ^ :=. - ? LL_^J 

p — 1 

and when ^— ^^ — <^ 0, by Formula D, we have 



J2^(l _ ^2)- 2 (^2 =r 






g-3 

- ^^-^-/^"(^ - 'T'^dz (DO. 
By the aid of the foregoing formulas, we are enabled to make 
the integration of ^m.^xco^.^xdx depend on that of expres- 
sions in which the exponents _2:? and q are numerically less than 
in the original function. From {k!) we have 

C • ^ n 1 sin.-p~^cccos.^ + ^a; 
/ sm.^oj cos.^o^ax = , • 

+ ^7" l^m.P-'^xGo^Sxdx (1) : 

and from (B^, 

sin.^ + ^ajcos.^-^a/' 



/ sin.-^cc co?,Jxdx z= 



p + q 



+ ^— — rsin.^cccos.^-2ccc?a;. (2). 

When q is positive, by the application of (2). 
Jsin.Pxcos.^xdx 

will finally depend on fsin.Pxdx, or on fsm.^xcos.xdx, 
according as q is even or odd. 



REDUCTION FORMULjE. 369 

By making g = 0, in (l), we have 

/. ^ 7 sin.^~^r:ccos.a; , p — lr- « 9 7 

and thus, if j9 be a positive integer, and even, fsin.^xdx will 
at last depend on fdx :=.x • and if p be a positive integer, and 
odd, the final integral to be found is Js'm.xdx z= — cos. a;. In 
the second case, that is, when q is odd, we have 

/ sin.^xcos.xdx z= I sin.^a;(isin.a; = — '—-- — . 

It is therefore always possible to find the integral 

fsin.Pxcos.^xdx 

when^ and q are entire and positive. 

Formulae 1 and 2 are inapplicable when p=^ — q; but in 
this case 

Jsin.^x cos.^xdx =:ft2in.Pxdx =zft8Ln.^~'^xtsin,'^xdx 
= rtan.^~2^(sec.2ic — l)dx 
—ftau.P~'^xdtiin.x —ftun.P-'-xdx 

This formula serves for the reduction of the exponent of 
tan.x; and the iutegratiou will at last depeiul on that of 

fdx := Xy or ft'du.xdx = I cos. .r, 
according as j9 is even or odd. 

221. In (1) of the preceding articlo, making (^--0, wo 
have 

C ' V J sin .''--^.c cos. a; , /> — 1 r • ., t 
j ^xwJ'xdx — — ^ \-^ -\ sm.'' -.n/.r; 

47 



370 INTEGRAL CALCULUS. 



and hence, whenp is even, 



cos.a; C . 1 p — 1 



p 1 ^^-2 



sm. 



(JP - 2)(i5 - 4) ^ 

I (j>-l)(ff-3)...3.1 

(i>-l)(p-3)...3.1 a; 
(i? — 2)(i> — 4)...4.2jj 



(1); 



and, when p is odd, 

/• « 7 COS. a; (.«! ,»— 1.^, 

p (^ ' p — 2 ' 

{p-l){p-Z)...2 } 
+ (i'-2)(p-4)...l5 (2). 

In like manner, by making p = in (2) of the preceding 
article, we should have formulas for 



I 



EXAMPLES. 

x'^dx _ x''-'^{2ax — x'^Y 



(2ax — ic^) n 



2n — l r x^'-^dx 

"/: 



n {2ax — x^y 

This will finally lead to 



/; 



dx . , X 
:=z ver. sm. ^ -• 



{2ax — x^Y a 

r x'^dx 



{2ax + x'^y 



x''~\2ax 4- x'^f 2n—l r x^'-^dx 

= ^ ^-— ^- aj 1; 

n n {2ax-\-x'^y 



EXAMPLES, 371 



and ultimately we should have (Ex. 7, p. 328) 

J z=z l(^x -\- a -\- >/2ax -\- x 

{2ax^x''f 

3. f(a'-x'Ydx = ''^'''~'^P" 

J ^ ' 271+1 



+ 

2n 



~1 ^'/(^' - x'-f-^dx. 



' 27^ + 

13 5 

If 71 be a fraction belonging to the series -, -; o--^ this 

2i u 2i 

process of reduction would at last lead to 

dx . , X 



/■ 



.2\^ tt 



(a^ — X' 
4. J(a2 _ x'^fdx == I (a' - a;^f 



o a 



5- /K + -T^-=^?-25-1«^/(»'-+-=)"-''^- 



+ 1 2n + 

1 ; 

2' 2' 2 



13 5 

"When n is any one of the fractions -, -, -...; the integral 



will finally depend on 

2n-3 1 ^ c?a; 

■^2(n- 1) a^ j(;^_a;')»-^' 

3 5 7 

Whon n is one of (ho scries of fractions , \ ...^ tho in- 
tegration will finally depend on that o( 



372 INTEGRAL CALCULUS, 

dx X 



I 



(a2 — a;2)2 a''{a^ - x^)^' 



by Formula D. 

„ r dx 1 X , 2 

T. J -, = 1 + - ^. 



/- 



x\a'' — x'-f in — l)a^x^-'' 

n — 2 1 r dx 



w — ^ i r 



n—1 a^" x^'-^a^ — x^f 
When n is even, the application of the formula will lead to 



/ 



dx _ {a'^-x'^f 

x^ip} — ic^)^ a^x 

and, when n is odd, the integral will depend on that of 

/dx 1 X la — {oP- — x'^Y 

x{a} — x'^Y a a-^{aP' — x'^y a x 

(Ex. 17, p. 332). 

2. 
c x'^dx _3a?"(a + 6:c)^ 3na ^ x'^-^dx 

{a-^hxf (372 + 2 )& Zn-^2h\a + hxf 

By the application of this formula, the exponent n will at 
last be reduced to zero, and the integration will depend on 
that of 

/ -^ = — {a^hx) . 

*" {a-\-hxy 2b 

^. r x'^dx . , ^ ,1(3^2 ^cix , 27a2) 
10. J -:={a + bx) ) 1 U 

{a + hxY LSh 20h' 405O 



EXAMPLES. 373 



11. r x'^dx _x''~^{a -\-'bx-\- cx"^) 



I 

[a-]- -f- cx^)- 

n — lac x^~'^dx 



1 — 

(a -|- 5 + cx'^Y nc 



n— L a r 

~~^ ^'' {a + hx + cx-'f 
2n— 1 h r x'^-^dx 

In c (a -\-hx -{- cx''-)'^ 
and, ultimately, we sliall have to find 

r xdx (a -{-hx -{- ex-) h r dx 

(a -]- bx -\- cx'^y c 2c (a -{- hx -{- cx-y 

dx 
but the integration of has been explained 

^a-\-bx + cx'^y 

(Ex. 1, p. 328). 

12. f "'^^^ , ^^+J(2-2^ + x-)^ 
'^ {2-2x-\-x''y 2 

+ ll^x-l + (2-2x + x'-^, 

13. Jx{2ax — x'')^dx ——^ {2ax - x'^f 

+ a / {2ax — x'^) dx. 



14. f~''x{2ax-x'^)^dx = '^~^' 

15. fx'\2ax - x-ydx = — ''^' {2ax - x'^f 



ni r ^ ox-} 






r — x-Ydx 



x-{2ax — X-) dx=: ■ - 
^ 8 



ui^xdx siu.u' .1,1 — sin..r 



-|Y /»siu.-x«a; sm.u. ^^ i 7 l — sin..r 

J cos.'*u3 ~~ 2cos.-a; ' i l+siu..c 



SECTION IV. 

GEOMETRICAL SIGNIFICATION AND PROPERTIES OF DEFINITE INTE- 
GRALS. — ANOTHER DEMONSTRATION OF TAYLOR'S THEOREM. — 
DEFINITE INTEGRALS IN WHICH ONE OF THE LIMITS BECOMES 
INFINITE. — DEFINITE INTEGRALS IN WHICH THE FUNCTION UN- 
DER THE SIGN J BECOMES INT^INITE. — DEFINITE INTEGRALS THAT 
BECOME INDETERMINATE. — INTEGRATION BY SERIES. 




222, Assume CPD to be the curve of which the equation, 

when referred to the rectangu- 
lar axes Oxj Oy, is y z=zf{x). 
It has been shown (Art. 164) 
that/(x)c?ic is the differential 
of the area of a segment of 
the curve terminated by a var 
riable ordinate ; and therefore 
Jf[x)dx is to be regarded as 
the expression for the area bounded by the curve, the axis of 
abscissse, and any two ordinates whatever. If this integral 
be taken between assigned values, a and 5, for x, the area will 
be limited in the direction of the axis of x by the ordinates cor- 
responding to these values of x. But the arbitrary constant 
may be determined by the condition that the area shall be 
nothing for x = OA = a : that is, shall be limited on one side 
by the fixed ordinate AG, while on the other side it is bounded 
by a variable ordinate corresponding to the variable abscissa 
OM=x. 

374 



DEFINITE INTEGRALS. 



375 



li ff{x)dx:=:-ip{x) -\- Cji\\Qr\,'bj the above condition, we 

should have 

i/;(a)-|-(7=0, (7= — V^(a); 
and 

Jf{x)dx = \p (x) — 1/; (a) 

is the expression for an indefinite area taking its origin from 
the fixed ordinate AG. 

When the other value o^ x, x^= OM' = &, is assigned, the 
integral becomes definite, and we have 



J a 



Definite integrals, when applied in the determination of the 
length of curves, the surfaces and volumes of solids, admit of 
a like interpretation. 

223, In Art. 192, it was shown that a definite integral was 
to be regarded as the limit of the sum of an infinite number 
of very small terms. 

To illustrate this proposition geometrically, let y =/(:c) be 
the equation of the curve CMD referred to rectangular co- 
ordinate axes, and suppose that, between the assumed limits 
x = a, x^^hj y increases continuously. 

HhQii f{x)^x measures the area 
of one of the small rectangles 
MP'.M'P''...] and if JS/(a;) A a; do- 
notes the sum of all these rectan- 
gles, the area ACDB included be- 
tween the curve, the axis of x^ and 
tlie ordinates y =i/(a), y zizj\h), 
will bo expressed by 

lim.2/(.r)Aa'.- C\/\.v)dx 



y 






3 
^ 


l' 




D 




r 


31 


X" 


M 







-^ 


^ J 


3 


P P 




1 


i'^ 



\i'[JA — v'{(i^. 



376 INTEGRAL CALCULUS. 

if \p {x) be the function of which f{x) is the differential co- 
efficient. 

224z. The order in which the h'mits of a definite integral 
are taken may be inverted, provided the sign of the result be 
changed for 

r f{x)dx='ip{h)-yp{a), 

*J a 

/"^ f{x)dx =: \p (a) — yj (b) : 
b 

r/{x)dx =- f f{x)dx. 

J b J a 

AlsO; if c be a value of x intermediate to the limits a and by 
we have 

fy{x)dx = xp{c) --ip(a), 

J'/{x)dx = xp{b) — v^(c), 

f /{x)dx=: ui{b) — ■w{a)] 

t/ a 

f^/{x)dx=z rf{x)dx+ f /{x)dx: 

and generally, if there be any number of values c, &, c" .,., 
between the values a and 5, it may be shown that 

r f{x)dx^^ V f{x)dx-\- V f{x)dx'-- -}- \ f{x)dx. 

225, Let/(ic), (^{x), be two functions of ic, so related that 
f{x) > <f(x) for all values of x from cc = a to x =ib ; then, 
taking y( a:) — <f{oc) for the differential co-efficient of another 
function, we should have 



J''J^/{x)-cr{oo)'^dx>0 



DEFINITE INTEGRALS. 377 

since the derivative /(ir) — g) (a;) is constantly positive be- 
tween the limits a and h, and the function I j f{x) — cp (x) i dx 
is an increasing function of x : hence 



f f(x)dx > r (p(x)dx. 

V a J a 



Also if (pi{x) is another function of x, such that q)i{x) >/(cc), 
for all values of x between the limits a and hj we should have 



//(x)dx<^ j cpi(x)dx: 
a V a 

therefore 

/(py^{x)dx y C /(x)dx > r q}{x)dx. 
a J a *J a 

When a given differential cannot be integrated, it is desira- 
ble, and sometimes possible, to find two other integrals be- 
tween which the required integral, at assigned limits, will be 
included. 



Example, t^ -. For values of x between and 



1, we have 



dx< j</ : 

0.5 < f~ '^^ ^ < 8in.-> ^ := 0.5:230. 

*^" [i-x'Y ^ 

220. Pemonstratiou of Taylor's Thooreni dopondiMit on 
the properties of definite integrals. 
The equation 

j\x -f h) -j\v) -^ f\r[x 4- // - f)df 

48 



378 INTEGRAL CALCULUS. 

is identically true ; but successive integration by parts gives 
V f'{x^li — t)dt — tf{x^li- t)-^ r tf'{x + h — t)dt, 

Jo »/ 

j'^ tf"{x + }l- t)dt =l^^f"(x+h-t) 

/.' &2'^"'^'' + Ji~t)dt = ~f"{x + h-t) 






Making t^^h in these equations, and then adding them 
member to member, we have 

f{x^h)=f{x) + hf'{x) 

If the function to be expanded, and also its differential co- 
efficients up to the order denoted by n-{-\j are finite, and 
continuous between the limits x and x + 7i, the residual term 

T~9 / f^''^^\x -\-li — t)dt may be replaced by 

and the expansion then agrees with that of Art. 61. 



DEFINITE INTEGRALS. 379 

227. In what precedes^ it has been supposed that the limits 

a and h of the definite integral \ f{x)dx were finite, and that 

the function f{x) was also finite, and continuous between 
these same limits. It may happen that one of the limits, 5, 
becomes infinite while the other is finite, the function re- 
maining finite and continuous. Then the value of the inte- 

/h 
f{x)dx when h is increased 
a 

without limit. This value may be finite, infinite, or indeter- 
minate. 

Example 1. f'"e~''dx. 

For the indefinite integral, we have 

Ce-'^dx — —e-''-\- C: 



• 




//- 


^c?x = 


:1- 


1 








/:- 


""dx- 


:1 - 


1 _ 


:1. 


Ex. 


'■ /:< 


I'^dx. 










The indefinite integral 


is 









Cc^^dx~e''-\- C: 
j e-^dx z=: e/^ — I z=i oo . 



Ex. a. r-, 

J ■'^■' + a' 



dx_ 
a'' 

dx 1 . . X 



Wo have f-T^ -^ = " t«in.- ' '" -f (7; 

./ x^ -\- a/ a a 

dx 1 , , 1 .7 

-^ - = tan. 'oo — . 



/JO 




380 INTEGRAL CALCULUS. 

/ao 
co^.xdx. 


In this case^ rcos.icc^o; = sin.cc + (7; and, taking the inte- 
gral between and the finite limit h, we have 



/ 



co^.xdx zzz sin. 6; 





but, when h becomes infinite, the value of sin.& will be inde- 
terminate, though confined within the limits and 1. 

The following investigation will sometimes enable us to 

decide whether the definite integral j f{x)dx is finite or 

infinite for &=:oo or&z=: — oo. 

First suppose that h is very great, but not infinite, and let 
c be a number comprised between a and 6 ; then (Art. 224) 

/f{x)dx=:: j /{x)dx -\- C f{x)dx. 
a *^ a *^ c 

Since /(o:) is finite, j /{x)dx is also finite; and it remains 

only to examine the value of j f(x)dx when h becomes in- 
finite. 

Put f{x^ ^= — ^, <]p(^) being a function that remains finite 

X 

for all values of x greater than c. If A denotes the greatest 
and B the least of the values of cp (x) for all values of x greater 
than c, we shall have 



DEFINITE INTEGRALS. 381 

Now, when n > 1, the second member of this last inequahty 

A 1 

for Z) = oo reduces to .: hence, in this case, we 

know that the integral I f{x)dx has a finite value. When 

•J a 

n <^ 1, we have 

Now, when 1 — n > 0, the second member of this inequality 

becomes infinite for & =: oo : hence f /(x)dx, and therefore 

/b 
f{x)dx is infinite for 6 =r oo . 
a 

If 71 = 1, then 

/vw''->^/;t=^'(')^ 

but 7( - J = cx) when Z> = oo : hence C'"/{x)dx = oo . 

Putting /(x) under the form -^^^-^ , ff(a;) being a function 

X 

that is finite for all values of x between — oo and some value 
less than h, it may bo shown in like manner that C /(x)dx 
is finite if n y 1, and infinite if 7i < 1 or n—l. 

Thus, if it bo possible to put f{x) under the form ?^ , ami 

X 

the condition imposed on (f{x) bo satisfied, wo can decide 

whether the integral / f(x)dx is linito or inlinito whou (>iio 

of its limits becomes -[- oo or — oo . 

228, Definite integrals in which the function under tho 
sign of integration bcconios inlinito botwoou or at the Hmits. 



382 INTEGRAL CALCULUS. 

The function f{x) may become infinite at one of the limits, 
&, of the integral \ f{x)dx; in which case the integral is 

defined as the limit of i f{x)dx when a is decreased with- 

ont limit. In like manner, \^ f{x) becomes infinite for x z= a, 

then / f(x)dx is the limit of f f{x)dx when a is indefi- 

nitely decreased. Finally, if /(c) = oo , c being comprised 
between a and h^ we should have 

r f{x)dxz=z\mx. V f{x)dx-^\\m. f f{x)dx, 

J a J a J c + 13 

when a and ^ are decreased without limit. Should there be 
more than one value of x for which /{x) becomes infinite be- 
tween the limits a and 5, we learn from what precedes how 

to define the integral j /(x)dx. 

229, It may sometimes be decided whether the integral 

f{x)dx is finite or infinite when f{x) is infinite at one of 

a 

fc(x) 
the limits. Suppose/(5) = oo , and let f(x) z= /^'^ ^ ; (p(x) 

( — X J 

being finite for x:=b and for all values of x <^ h, and n being 
>0. 
If c be a number comprised between a and h, we have 

f f{x)dx= rf{x)dx-\- f^f{x)dx. 
Now r''/(a;)(ia? is finite; and hence T f{x)dx^']S\. be finite 

J a J a 

or infinite according as \ f{x)dx is finite or infinite. 

Denote by A the greatest and by B the least of the values 
of (^{x) for values of x included between c and h. If ?^ < 1, 



DEFINITE INTEGRALS. 383 

we shall have, for such values of Xy f{x) < t^ ^,- : and 

therefore 

When a converges towards 0, the second member of this in- 

A 
equality converges towards the finite value ^ (h — cf~^\ 

hence, in this case, the value of lim. / f[x)dxj and there- 
fore of / f{x)dxy is finite. 

J a 

But if 71 > 1, the proposed integral is infinite; for, since 
•^(^) > {I _ r^Y ; we have 

J j[x)ax^j^ ^b-xf- 71-1 la"-' (Z,_c)«-M" 

and it is evident, that, when a becomes 0, the second member 
of this inequality becomes infinite : hence, under the supposi- 
tion, j /(x)dx is infinite. 

In like manner, when n =^1, we have f{x) ^ ^ ; and 

therefore 

h — c 

But Bl becomes infinite Avhen a vanislies : hence 

a 

//{x)dx, and thorolbro / J\x)dx, is tlion infinite. 
c *' a 

Fdx 



Example 1 



/ 



\/2b^ ^bx — x' 

P being a liinction ut" x that romalns finito lor all linito 



384 



INTEGRAL CALCULUS. 



values of Xy and a and h being two positive quantities, we 

have 

P ^ P 1 ^ ^{x) 

sJ^lV' — hx — x'- ~ \/W^x >/h^x {h — x)j 



by putting 



=:cp{x). 



s/2h^x 

Since the exponent oi h — x is less than 1, it follows from 
the rule just established that the proposed integral has a finite 
value. 

dx -nrx 1 r dx 

X 



Ex. 2 



/I dx 
VT^ 



"We ha^' 



-/ 



Vf 



— 2\/l 



a;; 



/ 



1-a 

Vl 



(ia; 



— 2^a; 



c?a^ 



^ V 1 — X 



dx 



The expression ,- is the differential of the area includ- 

V 1 — i^ 

ed between the axis of x^ and the curve having y =: .. 

V 1 — ^ 
for its equation. 

This curve has two asymptotes ; the one the axis of x, 

and the other a parallel to the axis of y, and at the distance 

+ 1 from it. It is seen from 

the figure the / . rep- 
resents the area bounded by AC^ 
AB, the curve, and its asymptote 
BD; and this area, although it 
extends indefinitely in the direc- 
tion of the asymptote BD, still 
has the finite value 2. 
230, A definite integral may become indeterminate, as is 
the case for 



2/ 


V 


D 


^ 




B =C 



/ Sm.xdx r=r cos. CO — COS. 0, 



INTEGRATION BY SERIES. 385 

since cos. x, when x is indefinitely increased, does not con- 
verge towards any determinate limit. 

For another example, take / — , in which a and h are 

J —a ^ 

any two positive quantities whatever. Since - becomes in- 

X 

finite for the value x z= 0, which is comprised between — a 
and + h, we put 

/+b dx ,. p-adx , ,. n+b dx 
— -^ lim. / — A- lim. / — ; 
_a ^ J -a ^ J +13 X 

a and ^ being numbers numerically less than a and b respec- 
tively, and the limits indicated being those answering to 
a = 0, ^ = 0. But 

/-adx ^ ' nb dx ,, ,_ 

■•• /::?+/;?=''-*+"«-'»='©+'e> 

Therefore ("^^ — —if- j + Vim. I \ 

The first term U-) in the value of this integral is deter- 
minate ; but, since the variables a and § are entirely inde- 



pendent of each other, the term lim. II j does not converge 

towards any fixed limit, and the integral is therefore indeter- 
minate. 

231, When the integral of Xdx is required, and A' can be 
developed into a converging series, 

X— Ui + u. + Us -\ + u„ -f r„ (1\ 

we shall have, after multiplying by dx, and integrating botwoon 
the limits a and h, 

r'A',?.r^. fu.dx + ru,.h-+..: + ru„d.v+ rr.,dr (•2). 

49 



386 INTEGRAL CALCULUS. 

If series (1) is converging for xz=^ a, x z^ h, and also for 
all values of x between a and h, Ave may assume r„ <^ a; 
a being less than any assigned quantity when n is taken suf- 
ficiently great. Whence 

/r^dx <C^ I adx, or i r^dx < a(6 — a). 
a *f a *f a 

Therefore \ r^dx will decrease without limit when n is in- 

J a 

creased without limit : whence the series 



/u^dx^ j Uo^dx -\- "• -\- r Un 
a *^ a ^ a 



dx 



is converging, and its sum is the expression for i Xdx. The 

fixed limit h may be replaced by the variable x, provided no 
values of x are admitted which fail to render series (1) con- 
verging. We should thus have 

I Xdx:=l Uidx-\-j iC2dx -\- " ' -\- j u^dx (3). 

232. Formula 3 of the last article still holds true for x :=h, 
even though the series ^^l + ^^2^~'^3~l~ ••' ? which is supposed 
converging for x <^ h, becomes diverging for x=:b, if, at the 
same time, Series 2 is converging. 

For, however small the quantity a may be, we have 

/' Xdx := j Uidx -\- j u^dx +•*• + / u^dx, 
a *^ a *^ a ^ a 

The two members of this equation are continuous functions 
of x, and are constantly equal ; hence their limits for a =: 
must be equal, and therefore 



/Xdx = r Uidx -{- j U2dx -{-'•' -j- j Un 



dx. 



INTEGRATION BY SERIES. 387 

If the series 

fix) =/(0) + a/'(0) + ^2/"(0) + ••• , 

to which the development of f{x) by Maclaurin's Formula 
gives rise, is converging, we shall have 

jf{x)dx = C+ a/(0) + -^2 /'(O) + ]^/"i^) +■■■; 

and, if it is wished that this integral should begin with ic = 0, 
C must be zero ; and we then have 

jy{x)dx = x/{0) + ^f{x) + Y^/'(o) + ... 

Example 1. f ^ — Z(l + x). 

By division, or by the Binomial Formula, we have 



l-\-X ' ' ' l-\-x 

,,^ . , X"^ , X^ X^ , X" nxX'^dx 

When X is numerically less than 1, positive or negative, the 
series 1 — x -{- x'^ — x^ .., is converging, and therefore so also 

X X X 

is the series ^ — ^c + ., between the same limits for 

A O 'i: 

x: hence, when i^<^i' 

It may bo shown by direct demonstration, that / \ , . 

converges towards as oi approaches c» . 

x" 
For, if X is positive, avo have i , . < -i'" •' therefore 

^r X"dx ^ ...r .r" + l 

Jo i+x<J/''^=n+l' 



388 INTEGRAL CALCULUS. 

Now, as n increases, — -j—,r approaches ; and consequently, 

in stopping with any term of the series, the error will be less 

than the following term, and will be additive or subtractive 

according as the last term taken is of an even or odd order. 

If X is negative, and a denotes a number greater than x, 

x^ x^ 
but less than 1, we have -^ < q : and therefore 



/, 



r^^ ^ {n-\-l){l-a] 



The limit of the second member of this inequality for n = oo 
is zero. In this case, the error is always numerically additive. 
When x=l, the series 1 — x -\- x"^ — x^ -}- -- - is no longer 

X X 

converging ; but the series x r + "o ' " ^^ (Art. 231), 

and will represent the value of 12 : hence 

We have ^ } ^^ — 1 — x'' -{- x' - x^ •- •zhx^'-'^T ^"^' 



.'5 rr^n 

tan.~^a;=r x 



n being an odd number, and positive. Integrating, and taking 
for tan.~^cc the least positive arc having x for the tangent, we 

find 

x^ _.x^ x"- nx cc" 

The series 1 — x^ -{- x^ — x^ ... ceases to be converging for 

X X 

x = l', but the series x — ^ -{- x '" ^^ ^^^^^ converging for 

o 

this value of x : hence 

4.-1 ^1 1,1 1 

tan. >x = ^ = l-- + g_^... 



INTEGRATION BY SERIES. 389 



Ex. 3. r'^-7=^=- = sin.-ia;. 
Jo Vl— ^' 

We have 



1 l+-^-^2+1^^4_,_l^|^6+... (1). 



VI — x' '2 '24 '246 

From this, by multiplying by dx, and integrating, we find 

. ■ , Ix' 1 Sx' 1 S 5 x' 

^^""^ = ^+2T + 24T + 246T-^ • 

a converging series when x^^^^i't since Series 1 is converging 
between these limits. 
The series 

ll-*- Ol-LO ..Loo n 

is not converging when x ^=1] but since, for x = 1 =z sin. 

the series 

, 1 a;3 , 1 3 0^5 , 1 3 5 oj^ 
^2 3 ^24 5^246 7 

is converging (Art. 231),^ we have 

^^.1 1,13 113 5 1 

2 ^2 3 "+"2 4 5 "^2 4 6 7^ 

A still more converging series is found by making 

1 . TT. 

^ = 2^'"'-G' 
whence 

"=' + ''. J +^^ ^ .... 

v) ^ ^ ^ , O •J 4: *j . i) 

* Space (loos not nlUnv tho proof of convoruom-o or divoruriioe wlioii those con- 
ditions are asserted relative to tho series involved in the last three exatn|des. (Sec 
Art. 68.) 



390 INTEGRAL CALCULUS. 

283. By integrating f(x)dx by parts, we have 
Jf[x)dx =: xf{x) — Jxf {x)dx, 

fx/'{x)dx = ^f>{x)-j ~f"{x)dx, 
Jxf"{x)dx ^ ^ /" {x) -J ^^ /'" {x)dx. 

The combination of these results gives 

ff{x)dx = x/ix) - ^/'(x) + ^/"(x) + ••• 

' 1.2.. .n -^ ^ ^ ' l.2...nJ -^ > ' 

This is the series of John BernouiUi, and may be advantar 
geously used in many cases : for example, if /(a;) be a rational 
algebraic function of {n — 1)*^ degree, /'"^ (a?) is 0, and the 
series will terminate ; or there may be cases when 

Jx''f^''\x)dx 
can be more readily found than r/(a?)c?x, or when only an 
approximate value of \f{x)dx may be required, and the in- 

tegral \ x'^f^^\x)dx may be small enough to be neglected 

without sensible error. 

234:, Assuming ff(x)dx = g) {x), and making a? == cc + A, 
we have, by Taylor^s Formula, 

^{x + h)-q,{x) = hq,'{x)+^--^q,"{x) + .-- (1). 

But, because Jf{x)dx = <f{x), we have 

f{x) = cp'{x), f{x) = cp^^{x), f'{x) = r'{x)' 



INTEGRATION BY SERIES. 391 

These values, substituted in (1), give 
^{x + K)-^{x)= \f(x) + ^ /' (X) + j^^ f"{x) + . . . 

In this series, making a? = a, }iz=:h — a^ and denoting by 
^1, Ac^^ A-^,..^^N\\dX f{x)^ f\x)^ f'' {x)..,^'hQQ,<dmQ under this 
supposition, then gi(a; + A) — (jp (^) becomes 

and we have 

This series enables us to find the approximate value of the 
f{x)dx when 5 — a is sufficiently small to 

a 

make the series converging. When this is not the case, or 
when the series does not converge rapidly enough for our 
purposes, put h — a^= na, and take the integral successively 
between the limits a and a -\- a, a -]- a and a -\- 2a, and so on, 
denoting the results by 

then (Art. 224) wc luive 

jyXx)d.r: = (J},+ C, + I>,+ ■■■)C( + {I1,+ C, + T>,+ ■■■)a' 

+ (/l, + C, + n, + ■■■),(■'+■■■. 
a series that may he made' to convor^'o as rapidly as we please 
by making a suilicieutly ^niall. 



SECTION V. 

GEOMETRICAL APPLICATIONS. 

QUADRATURE OF PLANE CURVES REFERRED TO RECTILINEAR CO- 
ORDINATES. QUADRATURE OF PLANE CURVES REFERRED TO 

POLAR CO-ORDINATES. 

235, The quadrature of a curve is the operation 
of finding the area bounded in whole or in part by the curve. 

If u denote th^ indefinite area limited by the curve, the 
axis of x^ and any two ordinates, it was found (Art. 164) that 

du := ydx ^=if{x)dx ; 
y z=.f(^x) being the equation of the curve referred to the rec- 



m 




tangular axis Ox, Oy, 

If it is desired to have the 
area limited on one side by the 
fixed ordinate CA, correspond- 
ing to the abscissa x =. OA == a 
-^ the integral must begin at a? == a; 
and we have 

u = f f(x)dx. 

Finally, if the area is to be limited on the other side by the 
ordinate BD, corresponding to a? = OB ^=. h, we have 

u =: area ACDB =:J f{x)dx. 

When the co-ordinate axes are oblique, making with each 
other the angle co, then 

u =z area ACDB = sin. co J f{x)dx. 

392 



QUADRATURE OF CURVES. 



393 



y\ 




M 



N B DC 



236, The definite integral is the limit of the sum, taken 
between assigned limits, of an infinite number of infinitely 
small areas (Art. 192). Observing that /(x) dx in equivalent 
to /(x) AXj if we suppose ax :=i dx to he positive, the element 
/{x) AX will have the sign of f(x). Consequently the integral 
will represent the difference between the sum of the segments 
situated above the axis of x and the sum of the segments 
situated below. 

If, for example, the ordinate 
changes, as in the figure, from 
positive to negative, and then 
from negative to positive, the 
area between the ordinates A 0, 
BD, will be 

f f{x)dx = ACL- MIN+ NBD ; 

and if OZ = /i, OiV=:/(j, the sum of these segments will be 
expressed by 

/h /» AT nb 

f[x)dx—\ f{x)dx-\- j f[x)dx. 

237. If y zz=.f{x) is the equation of the curve CM, and 
y^ =z \l){x) that of the curve C'3I', «t m 

and the area bounded by these 
curves and the ordinates AC, 
BM, corresponding to ic = a, 
ic = 6, is required, wo have 

Area C C 3131' = f J\x)dx - f xi\x)da 

50 




394 _ INTEGRAL CALCULUS, - 

EXAMPLES. 

Example 1. The family of parabolas is represented by the 
equation ?/" ■=^jpx'^^ m and n being positive. We have 

1 m 

and 

wnicn may be written 

71 L !^ n 

u = — j — p"" x"" x= — , — xy. 
m-\- n-'^ m -\- n ^ 



2sr 


M^ 


c 







K 


P ^ 



But xy measures the area of the rec- 
tangle 0P3INj contained by the co-ordi- 
nates of the point M. Hence, from the 
above formula, we have 

0PM: 0P3IN\\ n:m-\-n, 
0PM \ OMNwnlm; 
that is, the arc of the parabola divides the rectangle con- 
structed on the co-ordinates of its extreme point into parts 
having the ratio of n : m. 

Reciprocally, the property just enunciated belongs to the 
parabolas alone ; for the proportion 

0P3I: OMN::n\m 
may be written 

u\xy — u\\ n\m. 

Hence (m -j- ^i) '^ = nxy^ and, by differentiation, we have 

{^m-\-n)du^:. nxdy -\- nydx ; 

or, since du = ydx^ 

mydx z= nxdy : 

whence m — =in — . 

X X 



QUADRATURE OF CURVES, 



395 



Integrating 

nly = mix + C, or ly"" = Ix"^ + (7, 

putting Ip for (7, we have 

ly^ z= Zj9X™ or y^ =ipx'^ 

for the general equation of the curves which possess the 

property in question. 

For the ordinary parabola in which ti = 2^ m = 1, we have 

2 
u — -xy, 

Ex. 2. The hyperbolas referred to their asymptotes are rep- 
resented by the equation x^y"^ zzzp^ 
m and n being entire and positive 
numbers. 

Assume the asymptotes to be rec- 
tangular, and let NCM be the branch 
of the curve situated in the angle 
xOy. 

Suppose ny rrij and let u — area AC3IF, OA = a, OP =x; 
then 



"3 




V 











I iL P :>c 



or 



^ ydx = J ^ JO " x" " dx^ 

u — p>'(x " — a "" V 

n — m-^ \ j 



As X increases, so also docs u^ or the area ACJIP; and x and 
u become infinite at the same time. If, however, wo suppose 
PM to be fixed, and a to decrease, the surface, Avhilo continu- 
ally increasing, will remain finite ; ami at tlio limit, wlion a -^ 0, 

'^ ~ Hence the surface PJIXL 



it reduces to - 

'}i — in 



p" x 



approaches a fixed hniit as the point N approaches tho 
asymptote Oy. 



396 INTEGRAL CALCULUS, 

This limit, which may be written — xy, bears to the 

rectangle PMBO the constant ratio o^ n to n — m: since, 

denoting this limit by u, we have 

n 

n — m : n :: XT/ : u=z — xy. 

^ n — m ^ 

The converse of this is also true ; that is, no curves, ex- 
cept those represented by the equation x'^ y"" ^= p, possess 
this property : for, from the preceding proportion, we have 
u{n — m)zz^ nxy^ which, differentiated, gives 
{n — m) du = nxdy + nydx ; 
from which, by substitution and reduction, we have 

dx dy 
— m — =:n -^. 
X y 

Integrating 

nly ^= — mix -\- (7, 

making C = Ijp, then ly^ = I -^ : hence x^y^ zi^p. 

When mz^n, the general equation takes the form xy = p, 
which is that of the equilateral hyperbola of the second 
degree ; and we have 

p 7 dx 

yz=^,ydx^p~, 

and therefore u = plx + G ^^pl- 

by making G =^pl-. "When j9 == 1 and a = l,we have u=.lx; 

Cfi 

and the area is then equal to the Napierian logarithm of the 
abscissa. 

Ex. 3. The equation of the circle, referred to its centre and 
rectangular axes, is 

ic^ -}- 2/^ — ^^ • .'.2/ — '^^^ — ^S* 



QUADRATURE OF CURVES. 



397 



and ydx = S^a^ — x^ dx is the differential of the area of a 
segment limited by the axis of y and ^^ 
an arbitrary ordinate PM, Denoting 
this area by u^ we have 







x^ dx. 



Hence (Ex. 2, p. 326) 




2^ zz= ^ ic Va'^ — a;'2 + 



. _ X 



2"^ ^ ^ ^ , 2 
From this we deduce the area of the sector 0B3f ; for 

1 



the area of the triangle OMF is measured by ^ a; ^/^^ 
which, subtracted from the expression for u^ gives 



sector 0B3I:=. — sin ~^ - = a ^r sin." 

2 a 2 



a. V, 



that is, the area of a circular sector is measured by its arc 
multiplied by one-half of the radius. 

Ex. 4. If a and h denote the semi-axes of an ellipse, the 
equation of the curve referred to its centre and axes is 



a2^2^Z)2^2__^2 52 



-.^^-Va- 



x\ 



Let u denote the area of a seg- 
ment bounded by the axis of y and 
any ordinate, as PM ; then 

h /••'• / 

Describe a circle on 2a as a diame- 
ter, and denote by u' the area of the 
segment i?i!/PO; then 




J 



\/a'' 



x'dx. 



398 INTEGRAL CALCULUS. 

-i ^ u h 
u^ : u :: 1 : -} or —,^=-* 
a u' a 

That iS; the segment of the elhpse is, to the segment of the 
circle which corresponds to the same abscissa, in the constant 
ratio of 5 to a ; and therefore, denoting the entire area of the 
elhpse by A, and that of the circle by A', we have 

AlA'whla; 
and, since A' = ^a^, it follows that 

A^=.- na^ = nab, 
a 

Hence the area of the ellipse is a mean proportional be- 
tween the areas of two circles, having for diameters, the one 
the transverse, and the other the conjugate, axis of the ellipse. 

The ordinates FM, FN, are to each other as a to h, and 
hence the triangles OFM, OFN^ are in the same proportion ; 

that is, 

OFN FN h ^ u h 

=■ - ] but — = 



0F3I FM a' u' a 
u - OFN h OCN >b 



1 or 



u' - 0PM a ^' OEM a' 
and thus the area of the eUiptical sector may be found in 
terms of the area of the corresponding circular sector. 

An ellipse may be divided into any number of equal sectors 
when we know how to effect this division in a circle. It 
would only be necessary to describe a circle on the major 
axis of the ellipse as a diameter, then divide the circle into 
the required number of equal sectors, and through the points 
in which the circumference is divided draw ordinates to the 
major axis of the ellipse. The sectors formed by joining the 
centre with the points in which these ordinates cut the ellipse 
will be equal. 



QUADRATURE OF CURVES. 399 

Ex. 5. The equation of hyperbola is a^y"^ — S^^- = — a'^h'^, 

or ?/ = - Vx'^ — 0?- ; and the area of ^ 
^ a 

the segment AMF is expressed by 
h 






a? dx- 



Hence (Ex. 6, p. 328) 




AMP^^-x\^x'^ 
2a 



ah ^ lx-\- \/x^" — a 



/x + Vx''-a'\ 



2 \ a 

Ex. 6. The differential equation of the cycloid (Art. 146) is 



dx 



\2r — y \2Ty — y'^ 



This integral may be found by Ex. 1, p. 370: the following, 
however, is a more simple process. 

Put NM— 2r — y:^%; then, denoting the area OLKM by 
w', we have 

u' — Jzdx =f{2r — 7j) dx ~ ^ ^2ry - if- dy ; 

observing that the limits between which these integrals are 
taken must correspond to 2 = 2r and 2 == 2>' — y. But 
^s/2ry — y'^- dy is evidently the expression for the area of the 
segment of a circle of which 11 
r is the radius ; the segment 
taking its origin at the ex- 
tremity of a diameter, and 
having y for its base. This 
segment is represented by 
ADB. The area OXOT/ takes its ono-Iu from OL, and tlie cir- 
cular segment from the points/, and both areas are /.oro when 



IsT 


C 




/^' 


■) 




/M 


"V 




P 


I 


L 


X 



400 INTEGRAL CALCULUS. 

?/ = : hence the constant of integration is 0, and we constantly 
have area OLN]}I:=z segment ADB. 

When y 1= 2r^ the segment becomes the semicircle -^.D (7, 

which is measured by -— . But 

•^ 2 

rectangle OLCA = OA X AC = nr X ^r = Inr'^'^ 

that is, the rectangle is twice the area of the generating cir- 
cle : hence 

3 
area semi-cycloid OMGA =: -nr'^, 

and therefore the area bounded by a single branch of the 
cycloid and its base is three times the area of the generating 
circle ; or, in other words, this area is three-fourths of the rec- 
tangle having for its base the circumference^ and its altitude 
the diameter, of the generating circle. 

Although the area of the cycloid may be said to be thus 
represented by a part of a rectangle, it is not a quadrable 
curve ; for the base of the rectangle cannot be accurately 
determined by geometrical processes. 

238. Quadrature of curves referred to polar co-ordinates. 

The differential of the area 

FBM (Art. 165) is du=l r'^dd: 

hence 

the limits of this integral being 
^T the values of d corresponding to 
the points B and M, 

Example 1. Applying this formula to the logarithmic spiral, 
of which the equation is r = ae'"^, we have 




QUADRATURE OF CURVES. 401 



a^ '• 



J 4m 4m 



2 
Put F3I =: r\ and in the formula 



make r == r ' : then 



J 

ri 



4m 4m 



and u = - — 
4m 




-.m\ J 

The figure supposes PA to be the initial position of the 
radius vector; that is, the position at which (9 = and 
r zzz PA = a, and also that d is positive when the motion of the 
radius vector is in the direction of the motion of the hands of 
a watch. Hence, when the generating point moves in the direc- 
tion from A towards B, d is negative. Let the motion take 
place in this direction from the fixed radius vector PR i= r; 
then, after an infinite number of revolutions, r^ becomes 0, 

and the expression for u reduces to u :=i .-. 

2. When the length of the radius vector of a spiral is pro- 
portional to the angle through which it has moved from its 
initial position, its extremity describes the spiral of Archi- 
medes. The equation of this spiral is r = aO; and hence 
r z=z a when 6 = 57^.29578 of the circumference of a circle to 
the radius 1. 

For this spiral, wo have 

u=~Jrhid = I J a' 0'' do =z^, a' ()'-{- C; 

and, if the area begins when = 0, 

(7=0, and u = ^ a'O'K When = 2.y, 
6 

4 

u = area PAB =z ^^ a'-jr^ is the area do- 

t) 

scribed by the radius vector during i\\c lirst revolution. In 

51 




402 INTEGRAL CALCULUS. 

the second revolution, the radius vector again describes this 
area, and also the area PBA'B' included between the first and 
second spires. Hence the area PBA'B' is measured by 

It is evident that during any, as the m*^, revolution, the 
radius vector describes the whole area out to the m*^ spire, 
and that, to find this area, the integral 

A J O 

must be taken between the limits (m — 1)2;? and 2m;r, which 
will give for this area denoted by u" 



6 



^a\27ty^m'-{m — iyL 

In like manner, we have for the entire area denoted by u'^ 
out to the (m — 1)*^ spire, 

u' = ^a'(27ty I (m - ly - (m - 2y]: 

u" - u' = ^a\27ty^m' — 2 {m- ly + {m - 2y]^, 

which is the expression for the area included between the 

(m — 1)*^ and the m^^ spires. 

If we suppose a = ^ , this formula becomes 

u"—u'==~27t\m'-2(m-iy + {m-2y\ 

m'-2(m-iy-\-(m — 2y . ,,^ 

o 

and in this, making w = 2, we find 27t for the area included 
between the 1^* and 2*^ spires. Hence the area included be- 




QUADRATURE OF CURVES. 403 

tween the {m — 1)*^ and m*^ spires is m — 1 times that 
included between the 1^* and 2"^ spires. 

239, The quadrature of curvilinear areas is sometimes 
facilitated by transforming rectilinear into polar co-ordinates. 

Take, for example, the folium of Descartes ^ which, referred 
to rectangular axes, is represent- 
ed by the equation 

x^ -\- y^ — f^^y = 0. 

This curve is composed of two 
branches, infinite in extent, which 
intersect at the origin of co-ordi- 
nates, and which have for a com- 
mon asymptote the straight line of which the equation is 

To determine the area of any portion of this curve in terms 
of the primitive co-ordinates, we must find what the integral of 
ydx becomes when in it the value of y derived from the equa- 
tion of the curve is substituted. This requires the solution of 
an equation of the third degree ; but if rectilinear be changed 
into polar co-ordinates, the pole being at 0, there will be but 
one value of the radius vector in any assumed direction ; for, 
the origin being a double point, two values of r, each equal to 
zero, must satisfy the polar equation of the curve, and the 
first member of this equation must bo divisible by r^ 

Ox being the polar axis, the transformed equation is 

r^(cos.=' d -\- sin.=' 0) — af sin. cos. ^ ^ : 

whence ^ a sin. co^. 

~ sin.'' -}- COS.'' (f 

For the area of the segment OMN, we liavo k. = / y-dfl^ 



404 INTEGRAL CALCULUS. 

■which, by substituting for r its value above, becomes 
^■~2J.(cos.3^-l-sin.3^)2 — 2'J„cos.^(9a 



(cos.^^ + sin.^^)' 



To effect this integration, put 

= ata.-n.2^ 

cos/ d 



l + tan.3^ = 2; „'. cZg — Stan.^ (9 ^^^ ^ , 




and hence 

tan.2 6 






(l + tan.3^)* 

+ (7; 



t^ =: — 



3(1 + tan.2^) 

g^ 1 

6" 1 + tan.3 ^ 



+ (7. 



The area beginning when ^ = 0, we have (7 = -^, and con- 
sequently 

a2 tan.3 



u 



6 1+ tan.3 ^ 



The entire area OML is found by making 6 = -^ in the 

2 

value of It, which then becomes -rj for then the fraction 

tan.^ d __ 
1 + tan.3 "" ' 



SECTION VI. 

RECTIFICATION OF PLANE CURVES. 

24:0. The rectification of a curve is the operation of find- 
ing its length, and the curve is said to be rectifiahle when this 
length can be represented by a straight line. 

Denoting by s the arc of a curve comprised between a fixed 
point and an arbitrary point {x, y), we have 

ds = Vdx' 4- dj' = dx\l + ^' (Art. 161) ; 
and, by integration, 



=fd.^ 



i + 'lt 



dx^' 

By means of the equation of the curve, ds may be expressed 
in terms of either x or y ; and, the integral being then taken 
between the assigned limits, we have the length of the curve. 

Example 1. The Common I^arahola. From the equa- 
tion y'^zzzlpx of the curve, we find ydy z:^ pdx, dx ^^ ' '^. 
This value of dx^ substituted in the difiercntial formula, gives 

whence, making the arc begin at the vortex of the parabola, 

= I ^f- +r + i' Kit + v'F+y) + c, 

by Ex. 5, page o-T. 

406 



406 



INTEGBAL CALCULUS. 



Since the integral is to be zero, for y = we have 







^P 



Ip+C: 



C 



_ P 



Ip. 



2 -.-.--- - 2 

By the substitution of this value of O, the formula becomes 



VF+F+^^f^-±4^- 



Ex. 2. The Ellipse, From the equation of the curve 



r2«,2 



+ 62^2 _ ^252^ 



we get 



hence 



dy 
dx 






-"4 + l-?=^^J^ 



b^x' 



a'^(a'^b^ — h'^^x^) 



or ds = dx 



a' — {a' 



')x' 



a\a' — x-') 



dx 



a^ — e'-x' 



\ a' 



aJ q} — 52 _ 
in which e = • is the eccentricity of the ellipse. 

Suppose the arc CN to be estimated from the vertex G 
of the minor axis ; then, to get the 
length of the arc CNA^ the integral 
of the expression for ds must be 
taken between the limits ic == and 
x^=^a; but all the values of x 
between and a will be given by 
ic =: a sin. g), the angle cp varying be- 




tween and 



The substitution in 



the value of ds of these values of x and its differential gives 



z:=a\/\ — e2 sin.2 g) c?g} ; 



RECTIFICATION OF CURVES, 407 

and therefore 

J 

This integral belongs to a class of functions for which we have 
no expression except under the sign of integration ; and, to find 
its approximate value, we must have recourse to a series. 
The Binomial Formula gives 

(1 — e^sin.^g))'' — 1 — -€2gin.2g, _ -e^^m}cp 

113„.„ 11363.8 
-^^-e«sm.«^-----eW«gp...: 

hence, for the arc CN^ we have 

s — ag) — - ae^ J ^ui?cfdqi — - ~ ae^ f sm^(pdq) 

113 

The integrals in the second member of this value of 5 may 
be found by applying Formula 1 of Art. 221. We should thus 

get, by taking all the integrals betw^een the limits and'-, 

^'^ , 7t I o7t 11,13.^ 

r2Wl-e'sm,\dcp = ^-r2e 2-04^ 2 42 



_1 1 3 gl 3 5 TT . 
2 4 6^ 2 I 6 2***' 

hence, for the arc CMd, wo have 

^=-2p-(2V-304^)-5(24G^7 

_l/1^.5IeA^^. 
7 \2 4 6 8 J S 

This is a converging series, and tlio more rajuillv so as e 
becomes less, or as a and b npproaeh eipiality. When the 



408 INTEGRAL CALCULUS. 

eccentricity is very small, it would be sufficient to compute 
but a few of the terms of the series. 

This value of the arc CNA may be found without using 
Formula 1 of Art. 221 ; for, assuming the first equation in that 

article, and taking the integral between the limits and - , 

''2i 



we have 
















S>^' 


."^Cfdcp: 


m — 


ir^ 


sin."- 


'''(fdtf. 




~ m 


J, 


In like manner, 














■n 

p sin.™- 


-^(pdq) z 


m — 
~ m — 


■2J, 


sin."'" 


-*(pd(f, 




p sin.- 


~^q:dq) : 


m — 


■5p 


sin."" 


-^(fd<f, 




m — 


■4J, 



I 



- 1 

' sin.2g;c?9 — 2 2 



Multiplying these equations member by member, there 
results 

(in — l)(m — 3)(m — 5)... 3.1 tt 



/ 



^ ^ m(m— 2)(m — 4)...4.2 2 



The values given by this, by making m equal to 2, 4, 6,..., 
successively substituted in the value of s, lead to the result 
before found. 

The angle g) is found by the following construction : On the 
major axis as a diameter describe a circle ; produce the ordi- 
nate PiV to meet the circumference at J/, and draw OM; then 

xzzzOPz:^ 03Icos.F03I^ asm. B 031: 
hence g) = angle BOM. 

Ex. 3. The Hyperhola, Assuming the equation 
a2y2_ 52^2 ^_ 52^2 



RECTIFICATION OF CURVES. . 409 

of this curve, and proceeding as in the case of the elhpse, we 
get 

To simplif)^, put s/a^ -)- 6^ = ae ; then 
ds z:^dx 



\ x'^ — a' 



Now, for one branch of the hyperbola, all admissible values 
of X are comprised between + a and + oo , and for the other 
branch such values are comprised between — a and — oo ; and 
it is evident that all of these values will be given by the 

equation x =: by making (p vary between and - for 

one branch, and between - and Tt for the other. 



Substituting this value of x, we have 

, asin.(jpc?9 

cos.'-^qp ' 



J , V« e^ — a^cos.^o) , ae L cos.-(7i , 

and as^= ^ ^ d^D = ^— 1 .^ dec : 

cos.-cp cos."q}\ e- ' 

whence (fig. Ex. 5, p. 399) 

Jq cos.-gi'\ e^ 

Developing the radical in tliis integral, we get 

/.?) 1 (^ 1 cos.^op 1 1 cos.-^qp 

s — ae I < 1 — ~ — .. . 

J , cos.\fl 2 c- 2 4 e' 

1.1.3.5...(2/i-3) cos.^"(r) , 

Xix.:2^i e^^ r^' 

1 a 

or 5 = ae tan. f — cf — 

2 ('. 

(?. r<p\y 1 ('os.'-(jn I 1 :i cos.if , ) . 

e J, 1-2 4 e' + 2 -t « «r + ••• j'^T- 

52 



410 



INTEGRAL CALCULUS. 



The integration now depends on that of expressions of the 
form cos."*g}C?g), and may be effected by the application of For- 

mula 1; Art. 221^ after changing in it x into - — go. 

Ex. 4. The Cycloid. The differential equation of this 
curve (Art. 146) is 



dx := dy 



y 



\2r-y 



7 or 



dy z= dx I 



2r — y 



y 



y) 'dy: 



In the formula ds =. \dx'^-\- dy^j replacing dx by its value, 
we get 

ds — V2r(2r 

s— — 2 V2r(2r — y)^C. 

If s be estimated from the origin to the right, we must 

have 





y 


1 


■i 


y 


^ 






V 




d j 


e: 










Jff>/I 








A-' 


rrcy<^ 


T^j 


\ 


D 








J 


1 


: ] 


Q 





and 



4r+(7; .-. (7=4r, 



5 = OP =: 4r — 2 V2/-(2r — 2/). 
In this, making y = 2r, we 
have 00' J the semi-arc of the 
cycloid, equal to 4r, and the 
whole arc therefore equal to 
8r, or four times the diameter of the generating circle. 

Estimating the arc from the vertex 0' to the left, then 
(7=:: 0, since at this point y = 2r; and we have 

0'P=-2\/2r(2r-2/). 

But \^2T{2r-y)z=:\/GNx GC = FG: 

hence arc O'F = 2 chord FG; that is, the length of the arc 
of a cycloid, estimated from the vertex, is twice the corre- 
sponding chord of the generating circle. 



SECTION VII. 

DOUBLE INTEGRATION. — TRIPLE INTEGRATION. 

24:1, Double Integrals are expressions involving two 
integrals with respect to different variables. Suppose it is 
required to find the value of u which will satisfy the equation 

, , =r g)(ic, y)j the variables x and y being independent. 
ayCiiOc 

This equation may be written 

d /du\ ^ ^ dv 

(J 'II 

by making v ^=- ^ - The function v must be such, that its 

differential co-eflScient with respect to y^ x being considered 
as constant, is equal (p{x,y). We therefore have 

v=zf(p{x,y)dy = -J^: 

hence u must be such a function of x and y that its differen- 
tial co-effiicicnt with respect to x, y being constant, is equal to 

f(p{oo,y)dy; and therefore 

The value of u is thus obtained by integrating the original 
expression with respect to y, and thou integrating the result 
with respect to x. 

The last cipiatiou is generally and more concisely written 

u —fj\y{x, y)dxdy, or u — ffq{x, y^d//d.v ; 

•111 



412 INTEGRAL CALCULUS. 

the first form indicating that the first integration is performed 
with respect to x, and the second integration with respect to 
y. The second form indicates that the order of integration is 
reversed. 

(a 11 (Jj IL 

24z2» It was shown (Art. 91) that ^— ,-=-,-,-, or that 
^ ' dxdy dyax^ 

these partial differential co-efficients were the same, in which- 
ever order, with respect to x and y, the differentiation is per- 
formed. We will now prove that the result of the integration 
in the one order can differ from that obtained in the other 
only by the sum of two arbitrary functions, the one of x, and 
the other of y. Let u^, u^^ be two functions of x and y^ either 

d %i 

of which satisfies the equation ;,— ^- = <^ {x, y) ; then 

(xx(xy 

d'^Ux d'^u^ _ ^ 
dxdy dxdy ' 

dv 
Now, -J- cannot be a function Oi x, otherwise its differential 

co-efficient with respect to x could not be ; but it may be 
any function of y. Hence we may put 

^=/(2/); ^yhQXiCQVz=zJf{y)dy + x{x), 
in which x{x) denotes an arbitrary function of x. Putting 
J f{y)dy :^'ip{y),'yp{y) being as arbitrary qls /(y), we have 

finally 

v=zui — u^ = 'ip{y)-^x{x), 

as it was proposed to prove. 



DOUBLE INTEGRATION. 413 

24i3* A double integral / I g:(x, y)dxdy is tlie limit of 

all the products of the form 9 (a;, y)Lxb,y between the limits 
of integration. Let (jp (x, y) be a function of x and y^ which 
remains finite and continuous for values of x between a and 6, 
and for values of y between a and /5. 

To abbreviate; put 9 (x, ^) = s. Now, if we suppose x to 
be constant while y varies between the limits a and /3, we 
have (Art. 192) 

I zdy = lim. I.z^y. 

Multiplying both members of this equation by A a;, and- sup- 
posing X to vary between the limits a and h while y remains 
constant, there results 

2'Acc I 2C?2/ = Zt^x lim. Zzt^y : 
hence lim.^Aicl zdy — Vim.^^Axlim.ZzAy — lim.Z^UzAXAy, 

zdy — I I zdxdy 
by the article above referred to : therefore 

J J 9(^'; y)dxdy — Um. j::i<:p{x, y)ixiy. 

Writers do not agree as to the notation for double integrals : 
some making the first sign f refer to the variable whoso dif- 
ferential comes first in the integral, while others make the 
first sign J refer to the other varial)K\ In wliat follows, the 
first sign f will relate to the variable whoso diiferontial is tirst 
written in the indicated integral. 



414 INTEGRAL CALCULUS, 

24:4i, In the last article, it was supposed that the variables 
X and y were independent. It is sometimes the case, how- 
ever, that the limits in the first integration are functions of 

the other variable. For example, let i I q){x, y)dxdy be 

the required integral in which a = x{x)j and ^ = '^{x) ; then 

/' cf{x,y)dxdy= / 1 (p{x,y)dxdy. 

Suppose F(x, y) to be the result obtained by integrating, 
first with respect to y, regarding x as constant ; then, for the 
integral between the assigned limits for y, we have 

F\x,v,{x)\-F\x,x{x)\, 
and finally 

fafxt) '^ ^'^' ^-^ '^'"^^ = j[{F\x,w{x)\-F\x,i (X) \ )dx. 

When the limits of a double integral are constant, it is im- 
material in what order, with respect to the variables, the 
integration is effected ; that is, a change in the order of in- 
tegration does not require a change in the values of the limits. 
But when the limits for one variable are functions of the other 
variable, and the order of integration is changed, a special 
investigation is necessary to determine what the new limits 
must be to preserve the equality of the results. A geometri- 
cal illustration of this will be given in. the next section. 

245. Triple Integration. Let it be required to de- 
termine a function u of the three independent variables x, y, z, 

d^ u 
which will satisfy the equation , , , = V. We may write 

d^u d d'^u TT 



dxdydz dz dxdy 

d^U 7 d d'^U 7 TTl 

or -^-^ — - dz =-y~ -y— 7- dz = Vdz 

axaydz dz dxdy 



TRIPLE INTEGRATION. 415 

hence by integration with respect to 2, regarding x and y as 
constant, 

_^^ — fVdzA- T" ' 
dxdy '' ' 

T" being an arbitrary function of x and y. Again : we have 

dxdy dy dx J 
d'^ lib d dvb 

dxdy'^y=-dy dx dy='^yS ^^^^ + ^" '^y' 

which, by integrating with respect to y^ x and s being con- 
stant, gives 

p^=fdyfVd. + r + S'; 

T' being an arbitrary function of x and y coming from jT" dy^ 

and 8' an arbitrary function of x and z. 

Finally 

uz=z C^^dx^ fdxfdyf Vdz + T -{- S+B; 

T, B, and S being arbitrary functions, — the first of x and y 

resulting from TT'cZx, the second of a; and z resulting from 

J S' dx, and the third of y and z. 

It is usual to write the differentials together after the last 
sign of integration : the above equation thus becomes 

u =^fff Vdxdydz -\-T-JrS^B. 

This example suffices to show the manner of passing from 
a differential co-efficient of any onkM- oi^ a function o( several 
variables back to the function itself AVlien the variables are 
independent of each other, as has boon here supposed, there 



416 INTEGRAL CALCULUS. 

is no dependence between the arbitrary functions T, 8, R ; 
but more commonly at the limits of the integral the variables 
are not independent of each other. For example, the limits of 
the integral with respect to z may correspond to 2; = F{x, y), 
z = Fi{x, y) ] those with respect to y, to y ^=if{x), yz=.f^(x)'^ 
and, finally, those with respect to a;, to a; = a, a; =: h. 

By a demonstration similar to that given in the case of a 
double integral (Art. 244), it may be shown that 

/dx \ dy I q)Cx, y,z)dz = \im. 2,2:1: AX Ay A z, 
a J a *^ 6 



SECTION VIII. 



QUADRATURE OF CURYED SURFACES. — CUBATURE OF SOLIDS. 



24:6. Let F{Xj ?/? ^) = ^ ^g the equation of any surface 
whatever, and take on this surface the point P, [x^y, z), and 
the adjacent point Q, {x -\- ^x, y + Ly, z -\- ^z). Project these 
points in P\ Q^, on the plane 
X, y, and construct the rec- 
tangle F^Q^ by drawing par- 
allels to the axes Ox, Oy. The 
lateral faces of the right prism 
of which P'Q^ is the base will 
intercept the element PQ of 
the curved surface. Denote 
by X the angle that the tangent 
plane to the surface at tlie 
point P makes with the plane 
{x, y). This plane is determined by the tangent lines drawn 
to the curves Pq, Pp, at the point P. The tangent line to tlie 

first curve makes with the axis of x an auii'lo of which 

dx 

is the tangcutj and the tangent lino to the second makes with 

the axis of ?/ an an<;'lo of whicli ," is the tanuvnt. Those 

are the angles which the traces o[ the plane of these two linos, 
that is, of the tangent plane to the surface at the point P on 
the planes (2, a-), {z,7/), make with the same axes. Now, from 

63 417 




418 INTEGRAL CALCULUS. 

propositions 1 and 3, chap, ix., Eobinson's "Analytical Geome- 
try/' we readily find, without regard to sign, 

1 



COS. A =: 






The rectangle P'Q' is measured by Lxt^y^ and is the pro- 
jection on the plane {x^ y) of the corresponding element of the 

tangent plane. This element is measured by j. hence, 

for the element of the tangent plane, we have 
^x^y ( /dzV fdz\^)h 

=: sec.lAXAy. 
Let ^S' denote any extent of the surface under considera- 
tion, and assume that the limit of the sum of the terms 
sec. I AX Ay, for all values of x and y between assigned limits, 
is the area of the surface ; then 

'^-//S' +(£)'+($)'!'**• 

If the surface is limited by two planes parallel to the plane 
(z, y) at the distances a; = a, a? = 5, and by the surfaces of 
two right cylinders whose bases are represented by the equa- 
tions y = 9^(^), y = 'V^(^)j we should have 

«=/:-/:,':;i'+(i)"+(l)T* 

and, when the cylindrical surfaces reduce to planes parallel to 
the plane (sx), q){x) and '^^{x) become constants c and e, and 
the formula reduces to 

-/:-/;K(£)"+(S)T* 



VOLUMES OF SOLIDS. 



419 



247. Area of Surfaces of Hevolution. If ?/ —f{x) 

be the equation of a curve referred to rectangular axes, the 
differential co-efficient of the area of the surface generated 
by the revolution of this curve about the axis of x has been 
found (Art. 167) to be 

dx 



S 






dij 
dx 



\ ydx. 



248* Volumes of Solids, Consider the volume bounded 
by the surface of which F{x, y, z) z=iO is the equation ; and 
through the point F, {x, y, z), in this surface, pass planes par- 
allel to the planes {z, x), (z, y) ; and also through the point 
Q, {x^^x, y + ^y, 2 + A2), 
adjacent to the point P, pass 
planes parallel to the same 
co-ordinate planes. These 
four planes are the lateral 
boundaries of a prismatic col- 
umn, having P'Q' for its base, 
and terminated above by the 
element PQ of the curved 
surface. The volume of this 
column is measured by zAx^y, 
when AX, A?/, are decreased without limit; and the volume 
bounded by any portion of the curved surface, the j^lano 
{x,y) and planes parallel to the planes (^.v, //\ (~, .r\, will be 
the limit of iho sum of a series oi' forms oi' which zixsy is 
the type. Denoting this volume bv /' wo have 




V= - 



\,vsy = f f::dxdy. 



420 INTEGRAL CALCULUS. 

From the equation F(^x, y, z) = 0, which is the equation of 
the surface, we have z ^^ (p(x, y). If we integrate first with 
respect to y, we get the sum of the columns forming a layer, 
included between two planes perpendicular to the axis of x; 
and hence the limits of integration with respect to y become 
functions of x, and we should have fzdy =/(ic) ; f{x) being, 
in fact, the area of the section of the solid made by a plane 
parallel to the plane {z, y). Thence, finally, V =■ Jf{x)dx. 

249, Volumes of Solids of Revolution. The differ- 
ential co-efficient of the volume generated by the revolution, 
about the axis of x, of the plane area bounded on the one 
side by the axis of x, and on the other by the curve having 
y =/(cc) for its equation, has been found (Art. 166) to be 

hence, by integration, 

F= 7t J y"- dx ^^ J f{x) dx. 

Here, as was the case at the end of the last article, f{x) = rty'^ 
is the area of a section of the solid made by a plane perpen- 
dicular to the axis of x ; and the integral is the expression for 
the sum of the elementary slices into which we may conceive 
the solid to be divided by such planes. 

APPLICATIONS. 

Example 1. Required the measure 
of the zone generated by the revolu- 
tion of the arc 3IM' of a circle about 
B\~ P YK X the diameter BA. The equation of the 
circle is a;^ _|_ y2 _. ^2^ Denoting the area of the zone by 8, 
if OP^a, OF z=z h, we shall have (Art. 247) 




EXAMPLES. 



421 



S^2.Jj^l + C 



\dx 



dx 



/6 I /y.2 /»6 

y h + '^^dx—lTt / Bdx 
a ^ y *^ a 

=z 27tR (b-a) = 27tB X PP^ 

To get the entire surface of the sphere, the integral must be 
taken between the limits x :=z ~ B, x =: R, which will give 

Ex. 2. Suppose the ellipse of which 
the arc B3IA is a quadrant to revolve b 
about its transverse axis : required the 
measure of the surface generated by 
the portion BM of this arc, beginning 




01 p a:x> 

at the extremity of the conjugate axis. We now have 



^2./;,Ji + mw 



s 



From the equation of the ellipse, a-y'' -\-h'x'^ z= a'-b'-^, v,'q 

dy b'^x ^ 

Q:et ' = — — ;r- ; whence 
dx a-y' 



L fcfy\' __ \/ay + i^^ _ h\/a'- (a^ - b'')x 



dy 

dx/ a^y a^y 

and finally, by making V^'^ — b' — ac, wc have 



J 



1 + 



dx 



ay 



therefore 

S 



,^ b /»•'• , 'l:jbc /•■'• \a'- „ , 



422 INTEGRAL CALCULUS. 

But (Ex. % p. 326) 

Ne^ 2 \e2 



1 a'^ . ex 

2 e"* a 



therefore 



«=?H?-+?--?) 



If, in this expression, we make ic i= a, and take twice the 
result, we get 

^-2;t52 + ^sin-ie 

for the entire surface of the prolate ellipsoid of revolution. 

Suppose, now, that a <^ &, or that the ellipse is revolved 
about its conjugate axis, and put \^h'^ — a'^ ^=:he ; then we 
shall have 






a^y 



But (Ex. 5, p. 327) 

f^ I a' , ~ 1 Hi} ; a^ / i~a^ ' 

Jo^5V+"^"=2N6^+"+2P7^V+^^^^ 
therefore 



*=?= 'J^+'-'+sSf +J^.+«') +« 



Since this integral should be zero, for ^ zzi: we have 

a'' be 
hence 

r 






J- 



a' 



CUBATURE OF SOLIDS. 423 

If in this we make x = a, and take twice the result, we shall 
have 



for the entire surface of the oblate ellipsoid of revolution. 
If we suppose a^b, and therefore e = 0, the second term 

in the last expression for S takes the form -; but, by the rule 

for the evaluation of indeterminate forms, we readily find 

^/ be + Va' + b'e' 



h 



lim. 



whence we have 4;ta^ for the surface of the sphere. 
Ex. 3. Cubature of the JEllipsoid of devolution. 

The equation of the ellipse, referred to its major axis and 

the tangent line at its vertex, is y^ = — (2ax — a;-) ; and there- 
fore, for the volume of the ellipsoid, we have (Art. 249) 

^^ Tth'^ C^ ,^ „, , ttV" I , x^ 

K = — - I (2ax — x-)ax=z — - { ax- 

66- J a- \ O 

To got the entire volume, we make x z= 2a ; and then 

7rb' 



TfbW 



^ a'^ ) = -nb'^a. 
o / h 



This is the volume of the prolate olHpsoid. To got that oi^ 

the oblate ellipsoid, a and h mu^t be intorchangod in iho last 

formula. Wo thus got, for the measure o( the out iro volumo, 

4 , . 

-Tta^^b ; from which it is soon that this volumo is groator than 

o 

the iirst. flaking a — h, the ellipsoid booomos a sphere, the 



424 INTEGRAL CALCULUS. 

4 

volume of which is expressed by -Tta^] and, for the volume 

o 

of a spherical segment of a single base, the expression is 

Tt 

-x^^a — X). 
o 

Ex. 4. Volume generated hy the devolution of a 
Cycloid about its Base. 

In the formula V=^f7zy'^dx, substitute for dx its value 

i/ctt/ 
dx = Y derived from the equation of the cycloid. 

and we have 

J {2ry-y''Y 
but (Ex. 1, page 370) 

f-l^ = - ^ (2^2/ - 2/^f + 5. r^^ , 

J{2ry — y''Y 3 3 J (2ry — y'^f 

r f-dy ^ ^_y^2ry-y^UKf ^''^ . , 
J{2ry-y^y 2 2 J (2ry — y^)' 

J(2ry-y'y J (2ry - y'f 

= — {2ry — y^)" + r ver.sin."' - • 
therefore, by substitution and_ reduction, 



J{2ry-y''y \3 6 2 

+ ^Trr^ver.sin."^ - + O. 
2 r 



EXAMPLES. 



425 



Taking this integral between the limits y =zO^ y ^=z2r^ and 
doubling the result, we have, for the entire volume generated 
b}^ the revolution of a single branch of the cycloid, 

Ex. 5. Volume of an Ellipsoid, Take for the co- 
ordinate axes the principal axes of the ellipsoid. The equation 

of its surface is then -^ -^t^~\ — ^ = 1. 
a^ b^ c^ 

The section PMM' of the ellipsoid made by a plane parallel 
to the plane ZOy, and at the dis- 
tance OP = X from the origin, has 
for its equation 



y 



2' _ X^ 



h^ ' c- a' 

The semi-axes of this section will 
be found by making in succession 
2 :z^ 0, 2/ = ; they are 




N a- \ a- 



hence the area of the section is 



^ a- a- 



and, for the volume of the segment included bet ween the 
planes ZOy and P313P, we have 

V= -—- / ((r — x-)dx = — ( a-x — ) • 
a- J a- \ n / 

To get the volume ol" half the i^lli[)soid. niaki^ in this i'onnula 



;h i;-ives V 
is measured bv ., rtahc 



Txabc; and lionco the entire ^•ohuno 



64 



426 



INTEGRAL CALCULUS, 



Ex. 6. The areas of surfaces and volumes of solids have 
thus far been found by single integration. As an example of 
double integration, let it be required to find the volume 
bounded by the surface determined by the equation xy ziz az, 
and by the four planes having for their equations 

x = x^, x=x,^, y — Vi, y — Vi. 
The expression for this volume is 

''=/:;/;;f**=r.w-='5(':-<) 



= J {^2 - ^1) {y-i — ^1) (2i + 2., + ^3 + 24). 

in which z^, 29; ^3? ^4? ^^^ the ordinates of the points in which 

the lateral edges of the volume considered pierce the surface 

xy =z az. 

Ex. 7. To illustrate triple integration geometrically, in the 

figure suppose planes to be 
passed perpendicular to the 
axis of z. Let two of these 
planes be at the distances z 
and z -\- AZ res23ectively from 
the origin of co-ordinates, 
cutting from the elementary 
column PQ^ a rectangular 
parellelopij^edon ah measured 
by AX Ay A z. This parallel- 
opipedon may be considered 

as an element of the whole volume V: hence 




V = J J Jdxdydz . 



EXAMPLES. 



427 



Required the portion of the volume of the right cylinder 
that is intercepted by the planes z =: xidJi.d, z ^zxtau.d^ ; the 
equation of the base of the cylinder being x^ + 2/^ — 2aa:; =z 0- 
Here the limits of the integral are z := xtoin.d, z zz:zxtsin.6^, 
2/ ^= — V2ax — x'^, ?/ z= -|- \^2ax — x'\ x=zO, x^:z2a: there- 
fore, denoting the values of y by — y^ + 2/u 

' / dxdydz 

*y — y^Jx tan. ^ 

/ _ (tan. d' — tan. 6) xdxdy 

/2« 

xs/2ax-^x''dx 


= (tan. ^' — tan./9);ta^. 
The base of this cylinder is a circle in the plane {x, y) tan- 
gent to the axis of y at the origin of co-ordinates ; and the 
secant planes pass through the 
origin, and are perpendicular 
to the plane (2, x). The re- 
quired volume is therefore the 
portion of the cylinder included 
between the sections OP, OF', 
It can be seen from this exam- 
ple why, as was observed in 
Art. 244, Avhen there is a rela- 
tion between the variables at 
the limits of an integral, the order of integration cannot be 
chano'cd without at the same time ascortaininir if it bo not 
necessary to make a corresponding change in the limiting 
values of the variables. In this case, after integrating Avith 
respect to r,', wo integrate with respect to //. taking the into- 

gral between the limits y — — {2ax — .r-)*, // .ir -|- ^lax — .r'-)* ; 




428 INTEGRAL CALCULUS. 

that is, the integral is considered as bounded by the circum- 
ference of a circle tangent to the axis of y at the origin ; but 
by what portion of the circumference is not specified until the 
limiting values of x are assigned. The integral with respect 
to X is then taken from ic = to cc == 2a, which thus embraces 
the whole circumference. 

But it is obvious, that, if the order of integration with respect 
to X and y be reversed, then, that the integral may embrace 
the whole base of the cylinder, the limits with respect to x 
must be xz=a — Va'^ — y'^ , x z=z a -{- V a^ — y"^ ] and those 
with respect to y must be y = — a, y — -f- a. We now have, 
denoting the limiting values o^ xhj Xi^ — x^, 

/a nx^ n X tan. Q ' 
I I dydxdz 

— a J -x^ J ;rtan. 

= / j J (tan. d^ — tan. d)xdydx 

/« 

(tan.^' - tan.^) Va^ _2/2^^ 

z=(tan.^' — tan.^)7ta3 (Ex. 2, page 326); 
which agrees with the first result. 

250. IPolar Formula. The polar equation of a plane 
curve being r ^= cp{d), if s denote the length of an arc of the 
curve estimated from a fixed point, the difi'erential co-efficient 
of this arc (Art. 163) is 



=/!'■+©?* <"^ 



or, by taking r as the independent variable 



-/I 



'd6\ 2 
.2 / \ j^iydr (2). 



( \dr/ 



POLAR FORMULAE. 429 

Example 1. Applying Formula 1 to the spiral of Archime- 
des, the equation of which is r = ad, we have 

s ^/{r'' + a^)^dO = a/(l + O'^fdd 

If the arc considered begins at the pole where 6 = 0, then 
C=0. 
Ex. 2. For the logarithmic spiral, we have r = ba^ , or 

- . ~ T dd c 

r = he"" by makins; a =z e"" : . • , 6 ^=z cl -, and --,— = - : whence, 
•^ ^ ' h' dr r 

by Formula 2, 

If the limits of the integral correspond to the radii vectores 
ro, rj, the length of the arc is 

s = v(i + c^)(n-'-o). 

Since r— is the expression for the tangent of the angle 

made by the radius vector of the curve at any point and the 

tangent line at that point, we have, calhng this angle a, 

ds 
tan.a = c; hence sec. a = a/(1 -|- c-), and --- == sec. a: there- 
at 

fore 5 = r sec. a + C, and the definite portion of the arc an- 
swering to ro, o\, is (rj — r^) sec. a. 

251, To find the length of a curve in terms of the radius 
vector and the perpendicular demittod from the jntlo io the 

tangent lino to the curve at anv i)oint, wo have cos. o: == 

ds 

(cor. Art. 103) : hence, \{ p denotes the length o^ tlie jHM-peii- 
dicular, 

r r ds r 



430 



INTEGRAL CALCULUS. 



therefore 



ds 



dr \^r' 






rdr 



■ p" J \ T" — p) 

252. The length of a curve may also be expressed in 
terms of the perpendicular and its inclination to the initial 
line. 

Let X and y be the co-ordinates of any point M of the curve, 

and denote by s the length 
of the curve included be- 
tween the fixed point A 
and the point M. From 
the origin let fall the per- 
pendicular OP upon the 
tangent to the curve at the 
T ^ point iltf, and make OP—io, 
MP = u, and the angle POx = d; then, from the figure, we 

readily find 

p z^ X cos. -{- y sin. d, 

u = X sin. d — y cos. d. 
Also we have 

dy ds 

— = — cot. d, -7- = — cosec. : 

dx dx 




therefore 
dp 



dx 



-, =i — X sm. 6 -\-y cos. 6 + cos. ^ ^- + sin. 
do ^^ ' do ^ 



dy 



But, since 6 is the independent variable, 



dx 



cot. 6 mav 



be written 



dy 

de 

dx 
do 



COS. 
sin. ■ 



whence 



dy ^ dx ^ 

sm. d~ + COS. ^ -- =:= : 
do do 



POLAR FORMULAE. 431 

dp 

~ = — X sm. d 4-y cos. d = — u, 
do ^^ ' 

c?V du , ^ . ^dx ^ dy 

—4 = ~ — — X COS. d — y sm, d — sm. d -j--\r cos. ^ — • 

dd^ do ^ do ' dd 

dti 
But, from -^ =r — cot. 0, we get 

dy ^ dx 

cos. (9 -— =: — cos.^<9 cosec. d — - : 
dd do 

dx dy . dx „ dx 

. • . — sm. d- — h cos. (9 -— =: — sm. — cos.^(9 cosec. ^ -7- 

dx /sin.2 ^ _|_ (3Qg 2 Q\ ^j. 



, . , — cosec. d , 

\ sm.^ / dd 

. ds . ds dx 

The equation -7- = — cosec. gives -r- = cosec. <9 — - ; hence, 
^ dx ^ dd dd' ^ 

hj substitution, 

d'^p . as 

-z—r = — X COS. ^ — -y sm. ^ + -7- 



therefore 



ds 

dp 

do 



'^in + fP^O, 



or s^u— JpdO. 

Taking the integral between the limits ^^0, ^1, <^07 ^n ^'o; '^1? 
being the corresponding vahios of s ami ?/, we have 



5i —S^-\-U^ - U,, =: f ' pdO. 



The sign oT to will be positive or negative aeeerding a^^ the 
angle POx = is greater or less than the aiiuie JlO.v. These 



432 



INTEGRAL CALCULUS. 



results may be used for several purposes^ the most important 
of which are, — 

First, To find the length of any portion of a curve, the 
equation of the curve being given. In this case, from the 



equation of the curve and the equation 



dy __ 
dx 



— cot.^, X and ?/, 



and therefore ^:) = ic cos. d -\- y sin. d, can be determined in 
terms of d ; and, by integration, s may be found from the 

equation s = -~^ + Jpdd. 

Second, To find a curve, the length of a portion of which 

shall represent a proposed integral. Here, if the integral be 

Jpdd, 2D being a function of 6, the equation of the curve is 

found by eliminating 6 between the equations 

#..^ ^ . „,-^ ^ , (^P 

do 

hich we get from the equations 



x^=:zp cos. — ^^ sin. (9, y ^=-p sin. <9 + -^ cos. (9, 



p :z=:X COS. d -\- y siu. 6, ;7^ = — X siu. d -\-y COS. 6. 

The proposed integral will then be represented by s 



dp 




APPLICATION. 

Let BMC be a quadrant 
of an ellipse of which the 
equation referred to its 
centre and axes is 

(^2^2 _|_ J2^2 __ ^2^2^ 

This equation, by making 
&2 — (2^(1 — e^), may be 
put under the form 
y2 :^ (1 _ e^) (a^ - ^2). 



POLAR FORMULJE. 433 

Make P Ox = ^. 
Then, from the properties of this curve, we have 

0^2^-' tan.2 pro =:cot.2^ = (1^1^^!)^'. 
x^ a^ — x^ 

From the last equation, we get 

sm.2(9 = ~ ^-;r? C0S.2 d = -^ —^ : 



a^ — e^x 



OP ^OT"" X C0S.2 d=:a -2 0-^; 



or OP=p=: c^^( ^2_g2^2 = « Vl- e'^ sin.2 ^. 

Therefore 

Oilf + i)/P z= 5 + r^ =r aJVl-e'^sin.^^ c^^. 

It is here supposed that the integral takes its origin at C, 
the vertex of the transverse axis. Now, if the point A be so 
taken that the angle BOA^^ d, it has been shown (Art. 240) 
that 

Arc BA — a fVl — e'^ sin:^ (J dd : 

CM-^MP^BA. 
Also we have 

j.j.p _ dp __ ae^ sin. d cos. 
~ ~ do ~ Vr^Hmfd ' 



and, X being the abscissa of the point iLT, 

dp 

do 



dp . 
xz=p COS. — ; sm. 



a(l— c-sm.-^)-cos.^ + 



(1 —c''sm:'0)^ (l — c'-siurO)^ 

65 



434 INTEGRAL CALCULUS. 

Therefore MP ■= e'^x sin. d ; and, x' being the abscissa of A, 
we have x' :=z a sin. : . • . MP == , and hence 



BA — CM— MP-- XX', 
a 

a result known as Fagnani's Theorem. 
From the values of x and x' ^ we get 



X'^zz^ 



2 

a^ — a*- sin.^ d a^ — x' 



i-e^sin.^^ e^'" 

a? 
which gives 

e^x'-x'- a'-ix" -\-x')^a^— 0, 
an equation which is symmetrical with respect to x and x' : 
hence, if we have 



e' 



we also have 



BA- CM— —XX', 
a 



BM— CA = -xx'. 
a 



258, Curves of Double Curvature, A curve of 
double curvature is one, three of the consecutive elements 
of which do not lie in the same plane. Such a curve must 
be referred to three co-ordinate axes, and requires for its 
expression two equations which represent the projections of 
the curve on two of the co-ordinate planes. 

Let the equations of the curve be 

y=f{x) (1), , = ^{x) (2); 

(1) being the equation of the projection on the plane {x, y), 
and (2) the equation of the projection on the plane (x, z). If 
X, T/j z, are the co-ordinates of a point of the curve, and 



CURVES OF DOUBLE CURVATURE, 435 

X -{- hx^y -\- ^y, 2 + A2, the co-ordiDates of an adjacent point, 
then, by the principles of solid geometry, the length of the 
chord connecting these points is 

Then, if s is the length of an arc of the curve estimated from 
a fixed point up to the point (x, y, 2), that of the arc from the 
same fixed point up to the point (r^; -|- a a;, y -\- Ly^ s-f-Ag) 
will be expressed by s + A5, We shall assume 

,. A5 

lim 



|(Aa.)2 + (Ay)' + (A!5)^p 



AS 



and therefore 



dx I ^ \dx) ^ \dx) ) 

-/l'+(S)'+©T- 

The two equations of the curve enable us to express -y^, -^, 

ax dx 

in terms o^ x ; and, by integrating, s will then be known in 
terms of x. 

Any one of the three variables may be taken as independ- 
ent; and the above formula may bo chauired into 



or 



=/5-(§J+(l)T'- 



436 INTEGRAL CALCULUS. 

When X, y^ and z are each a known function of an auxiliary- 
variable, t, as may be the case, then 

dy dz 

dy _ dt_ dz _ dt . 
dx ~ dx dx~ dx 



dt dt 



and we may have 



or 



or 



-/1(S)*+(S)'+(S)T- 



254,, To convert the formulae of the last article into polar 
formulae, take the "pole at the origin of co-ordinates, and denote 
by d the angle that the radius vector makes with the axis of 
2, and by g) the angle that its projection on the plane (cc, y) 
makes with the axis of x ; then we have the relations 
x^^r sin. 6 cos. (p, y ^r sin. d sin. gy, s = r cos. 6. 

These three equations, together with the two equations of 
the curve, make five between which we may conceive r and qo 
to be eliminated, leaving three equations between cc, ?/, 2, and 
6 : hence, x, y^ and z may be regarded as known functions of d. 

Therefore 



d nr ci ^ Cm CO 

—- r= sin. d COS. w r sin. 6 sin. o) — - + r cos. d cos. (p 

do ^ dd ^ dd^ ^ 



dy . . dr . dcp 

-— sm. sm. CD —- -\-r sm. 6 cos. qi -^ 4- r cos. 6 sin. cp, 

do ^ dd^ ^ dd^ ^' 

dz dr 

— - =: cos. d — r sm. 6 : 

dd dd 



POLAR FORMULAE. 



437 



dxV , /dyV fdzV 

^J + W+W ^ 

d(f,V 
do) 



= (|\V,.,i„. 



and 



-/)'■+©' 



-|- r^ sin.^ 



\ddj 
do 




do 



which, by changing the independent variable, may become 

/rlR\ 2 /rl^\ 2 ^ 1 



or 



=/'■©'+ 



-y- I -f- r^ sm. ( 
dcf/ 




23S. l^olar Formula for I*lane Areas. In the 

curve B3IQ of which the polar 
equation is r — cp(d), let r, (9, be 
the co-ordinates of the point 31, and 
denote by A the area bounded by 
the curve, the radius vector FB 
drawn to the fixed point B, and the ^^ 
radius vector F3L Then (Art. 165; 
dA 1 ( , )2 

Lot 'i/'(^-') bo the function ha\'ing cp^f)) for its ditlVroutiul 
co-efficient; then A :z= if > {())-{- C : and if ^-/i, A.,, donoto the 
areas corresponding to the values 0^, ().>, oi' the vectorial angle, 
wo have 

-^=^V•(^^) + <^, A,= U'{0.;)-\-C: 

1 /V'. 



lf\n^» 



d). 



A 






438 



INTEGRAL CALCULUS. 




Example 1. For the parabola, when the 
pole is at the focus, and the variable angle, 
measured from the axis, begins at the ver- 
tex, we have x:=p — r cos. 6^ y =zr sin. ; 
from which, and the equation y"^ = 2px of 
the curve, we get 

do 



1 + COS.^ 2 2^' ' ' 8 , 4 

' COS/- I COS.*- 



■••-.--.=?(*-i-'*4)+S('"4'— -'I)- 

9 9 9 

.., , . ^ . 7t , n , P^ P^ P^ 

Makmg 6i = 0, d^ =-, we have for the area -r + t^? or —' 



Ex. 2. The equation of the logarithmic spiral being r ^= be' 
we find 

A = - fb^e-^dd^ — ec + (7, 
2 J 4 






C / 2 2 

e '^ I — -(r — r 




2S6. A polar 
formula involving 
double integration 
may also be con- 
structed for plane 
areas. Suppose the 
area included be- 
tween the curves 
B3IU, bme, and the 



POLAR FORMULJE. 439 

radii vectores PB, PE, is required. Divide the area up into 
curvilinear quadrilaterals by drawing a series of radii vectores, 
and describing a series of circles with the pole as a centre. 

Let no be one of these quadrilaterals, and denote the co-ordi- 
nates of n by r, d; and of o hj r -\- ^r , 6 -{- ^^. Now, the area 
no is the difference between two circular sectors ; and the 

accurate expression for this difference is r^r^^-{- -{^ryAd^ 

A 

the ratio of the second term of which to the first is 

-(^TY^^ 

r^r^^ 2r 
This ratio diminishes as ^r diminishes, and vanishes when 
Ar — : therefore we may take rArAd as the expression for 
the elementary area, since, in comparison with it, the neg- 
lected term - (Ar)^ Ad ultimately vanishes. 

257* In the last article, it was shown that tavaO might be 
taken as the expression for the polar element of a plane area. 
If we suppose this area to be the section of a solid by the 
plane {x, y), the column perpendicular to this plane, standing 
on the clement rArAd as a base, may be regarded as an ele- 
ment of the solid. The volume of this column is measured 
by zrArAd ; and therefore, for the volume V of the solid, we 
have V— J JzrdrdO. 

The value of z as a function of r and will bo given bv 
the equation of the surface bounding the solid. 

Example. Required the measure of the volume lunindod 
by the plane (.r, y), and the surfaces having 

x'^y''-az-^{) (1), x''-\-y"---lKv = ^'1), 
for their respect ivo oipiations. 



440 INTEGRAL CALCULUS. 

Denoting tlie polar co-ordinates of a point in the plane 
(07, y) by r and d, d being measured from the axis of x, we 
have 

ic = rcos.^, y^r^m.d (3); 
therefore x'^ -\- y'^^ =z r^, which, combined with (1), gives 

r"^ z=^ az : . • , 2 = — . 
a 

From (2) and (3) we find r = 2hcos.d : hence, for this exam- 
ple, we have 

V= JJzrdrdd = J J— drdd. 

To embrace the entire volume comprised between the sur- 
faces indicated, the integral must first be taken, with respect 
to r, between the limits r == 0, r == 26 cos. /9, since is assumed 
as the independent variable ; and then the integral of the 

result must be taken between the limits 0=^-^1 =: — - . 
Thus 

I2&C0S. 




46^ 



r^\ \ -drdd=z—\ cos.'ddd 



cos^dddz=l-7t (Art. 221). 



258 • Suppose the polar element r^r^^ of a plane area to 
revolve through the angle 27t) around the fixed line from 
which the angle d is estimated. A solid ring will thus be 
generated, the measure of which is 2Ttr^m.^r^r^^ ; since, in 
this revolution, the point whose polar co-ordinates are r, 6^ 



POLAR FORMULAS. 441 

will describe a circumference having rsin.d for its radius. 
Denote by 9 the angle which the plane of the generating 
element in any position makes with its initial position ; then 
cp -\- Aqj will be the angle which the element in its consecutive 
position makes with the initial plane. That part of the whole 
solid ring which is included between the generating element 
in these two positions is measured by 

(cp -\- Aq))r'^ sin. Oat Ad — cpr'^ sin.dArAO := r^ sin.dArAdAcf. 

This may be assumed as the expression, in terms of polar 
co-ordinates, for an element of the solid : hence, for the vol- 
ume V of the whole solid, we have 



V— f f f r^ sin. ddrdddqj, 



in which the limits of integration must be so determined from 
imposed conditions, that the integral may embrace the entire 
solid to be found. 

Example. Required the volume of a tri-rectangular pyra- 
mid in a sphere. Integrating the above formula, with respect 
to r, between the limits r = 0, r = a, a being the radius of 
the sphere, we find 

V— r f fr' sin JIdrdOdx ~ f T^ sin. OdOdif. 

Now, a'' s\n.OAOA(f is an oleniont of the splierii-al siirtaoe : 

and sm.OAOAcp is tlioroloro the expression lor an oUMnontary 

spherical pyramid having a'^ su\.f)Afisi-p for its base. r>v this 
first integration, therefore, the eliMnont of the vohmu^ has 
changed from au ehMueut of the solid ring, gonoratod by the 
revolution oi' vAriO, to ;ni elomontary spherical pyramid. 

6G 



442 INTEGRAL CALCULUS. 

Integrating next, with respect to 6^ between the limits 

we have 

since 

fsinMd = — COS. d: . * . T sin. Odd —I. 

2 

By this second integration, the elementary volume has 
become a semi-ungula, or a spherical pyramid, having a bi- 
rectangular triangle for its base ; the vertical angle of the 
triangle being Aqp. 

We finally integrate, with respect to qp, from gj = to 9 = ^; 

and get for our result 



SECTION IX. 

DIFFERENTIATION AND INTEGRATION UNDER THE SIGN /. — EULE- 
RIAN INTEGRALS. — DETERMINATION OF DEFINITE INTEGRALS 
BY DIFFERENTIATION, AND BY INTEGRATION UNDER THE SIGN /. 

259, Whatever function of x^ /(^) may be, there exists 
another function, Qp{x), of x, such that <f' (^x) ^=zf{x)] and 

therefore \ f{x)dx ^:^ (pi^x) -\- C (Art. 191), G being an arbi- 
trary constant. 

Denoting by 1^ the integral of /{x)dx, taken between the 
limits a and h, we have 



= f /{x)dx = cf{h) — cp{a). 



The definite integral u is independent of x, but is a func- 
tion of the limits a and b; and its differential co-eflScient witli 
respect to either of these limits may be obtained without 
effecting the integration. For, since 

ic =: qih) — (]r(a), 
wo have 

du , dii 

and, because (f^{x) ^=/[x), 

die . , (/// 

du z=z/\l))db—j\a)da, 

443 



444 



INTEGRAL CALCULUS. 



ILLUSTRATION. 

Let y zzzf{x) be the equation of the curve if JV referred to 
the rectangular axes Ox, Oy. If a 
and h are the abscissae of the points 

f{x)dx will repre- 

a 

sent the area AMNB. Give to a and 
h the increments 




A A' 



then 



B B' X 


AA' — ^a, BB' 


= Ab; 


Au AMM'A' 

Aa~ A A ' 


Au BNN'B' 
Ab ~ BB' ' 





The definite area AMNB is obviously a decreasing func- 
tion of the first limit a, and an increasing function of the 
second limit h : therefore 



hm.-=hm.^^^^-/(a), 



AU ,. BNN'B' ,,,, 
urn. — ^ = Inn. — ^.^^^ — =/(6). 



Ab BB' 

Regarding the areas AMM'A' , BNN'B', as elementary,. we 
see that the total increment of the area AMNB is the diifer- 
ence of the increments that it receives at the limits. 

260» SupjDose /(x) to contain a quantity, t, independent of 

X, and that the differential co-efficient of / /(x)dx with re- 
spect to t is reouired. Replacing _/(cc) by /{x, t), we have 

/(x, t)dx, 

a 



If the limits a and b are independent of / we have by giv- 



DIFFERENTIATION UNDER THE SIGN f. 445 

ing to t the increment Lt {au being the corresponding incre- 
ment of u), 

/b nb 

f{x,t-\-At)dx-~i f{x,t)dx 
a *^ a ' 

'^f^(/(x,t + At)-f{x,t)yx: 

At J ^ At 

Now, by Art. 15, we may write 

f(x,t + At)-f(x,t) _df{x,t) 

At ~ dt "^^^ 

in which / is a quantity that vanishes when A^ vanishes. De- 
noting by y^ the greatest of the values of 7, we have, gener- 

ally, 

/ ydx <:^{h-a)f', 

*J a 

and, when neither a nor h is infinite, (b — GL)y' ) and therefore 
ydx^ will ultimately vanish : 

At dt J „ dt 

APPLICATION. 

Resuming the formula 

just established, suppose g) (a;, ^) to bo the function of which 
/(a;, t) is the difi'erential co-cfiicient with respect to x, and 

\p{x,t) to bo the function of which ^' ^y ' is the dllVerontiul 

co-efficient with respect to x; then (I) becomes 



446 INTEGRAL CALCULUS. 

If y*(rr, t) and a are both independent of h, (2) may be written 
^ll) + C = Hi,t) (3); 

C denoting the sum of the terms which are independent of h. 
Since we may give to h in (3) any value we please, replace h 
by X ; then (3) becomes 

n,{x,t) = ^^f^ + C (4). 

Dropping the constant C, which may be restored when neces- 
sary^ and putting for the other terms of (4) their equivalents, 
we have 

1 dx 

Example. Let f{x, t) = jj-^i ^ ^^^^ /(^; 0^-^ = I+i^^ •' 

I f(x,t)dx =^ I — - — - tan.~^^a;, 

and — f/(x,t)dx = --(-t8.n-Hx\= f '^li^ dx 



=Ilt (iT^) ''^ = -/(if^T ^^- 



/dx 
^ — r^ , we findj by differentia- 

tion, that of the more complex integral / — 2 ^■^• 

f{x, t)dx, both a and h 

a 

are functions of /, then -j- will consist of three terms ; since 

at 

in this case, to obtain the total differential of u, we must dif- 
ferentiate it with respect to t, and also with respect to both 



DIFFERENTIATION lJNJ)Eli THE SIGN f. 



447 



a and h regarded as functions of t, and take the sum of the 
results. Thus we should have 
du 






diL dh dii da 
db dt da dt 



;^,.+/(M)§-/(a,4j(Art.259). 

Under the above suppositions, the second and higher differ- 
ential co-efficients of u with respect to t may be found. Thus, 
by differentiating each of the terms of the last formula with 
respect to t^ we get 
d'^u r'dy(x,t) 



W' ~J a 



di' 



'-dx 



d'-h df{h,t) fdbV df{b, t) dh 



dt dt 



. d'^a df(a, t) /daV- df{a, t) da 

--^^""^ ^^W c^a~ VcU) " ''dt'^ Tt 



ILLUSTRATION. 

Let y =y*(a:, t) be the equation of the curve CD referred to 
the rectangular axes Ox^ Oy, and 
y =f{^x^ t -{- ^t) that of the curve ^ 
EF. Put 

031 =z a, ON=h, 

/{x, t) dx denotes 

a 

the area 3INDC, and u -\- \u the 
area M'N'FE: 

Mi^ EE'F'F + DNN'F' - JLV'E' C, 

MJFE'C 




>1 M 



Au _ EEFl^' DNN'F 

At~' At At 



At 



448 INTEGRAL CALCULUS. 



A U 

It is plain that the first term in this value of — is the ratio 

A^ 

of A^ to the increment of the area due to the change from 
the curve CD to the curve EF. The limit of this ratio is the 
limit of f f{^,t + ^t)-f{x,t) ^^^ g^^ ^^^^^ ^^^ j.^.^ ^^ ^^^ 

J a At 

second term is the limit of f{h, t) —- , and the limit of the third 
term is the limit of /(a, t) — : hence 



du 
~dt 



r^ df(x, t) J , ^/i .^ dh „, ,. da 



which agrees with first formula established in this article. 

262, An indefinite integral may also be differentiated with 
respect to a variable contained in the function under the sign 
of integration which is independent of the variable to which 
the integration refers. 

Let the integral be u =if/(^x, t)dx, t being independent of 
x: then, without impairing the generality of this integral, we 
may write 

/x 
f{x, t)dx -\-\p{t)'^ 
a 

\p{t) being an arbitrary function of t. Differentiating with 
respect to t, t not depending on a, we have (Art. 260) 



s=/:%^'^-+^'(') 



but, since '^\t) is a constant with respect to x, it may be 

/dfix t) 
'^ ^j^ ' dx ; and 

hence the last equation may be written 

du_ r d/(x, t) 
dt-J —df'^''' 



INTEGRA TION UNDER THE SIGN f. 



449 



and we have only to differentiate the function under the sign f 
with respect to t. 
263, Integration under the Sign of Integra- 

/b 
f{x,y)dx as the differ- 
a 

ential co-efficient of y, and integrating, we have 

jdyj f(x,y)dx 

for our result; and it is proposed to prove that this result is 
the same in whichever order with respect to x and y the inte- 
grations are performed ; that is, we shall have 

/"^^y/ f{^iy)dx=j dxjf{x,tj)dy. 



For 



/ dx /{x,y)dy= I dx - 



dy 



dy 

/b 
f{x,y)dx. 
a 

Integrating the two members of this equation with respect to 
y, we get 

J dxj/{x, y) dy — J^^l/J f{^, V) ^^^' / 

and, if the limiting values of y arc c and d, we shall have 

/b nd 

dxj '/{x,y)dy 

= f^h/ f JV,y)dx, 

The figure gives the geometri- 
cal interpretation of this formula. 
Either member n^presents (lie vol- 
ume AC iucliuled bolwoen llio 

piano (.r,//), the surface A' W C D' having - ~f^x,y\ for its 
67 




450 INTEGRAL CALCULUS. 

equation, and the planes whose equations are xz=za, x :=h^ 
y — c, y = d. 

Example 1. Find the form of the function (p(x) such that 
the area included between the curve y=^(p[x), the axis of x, 
and the ordinates y ^=0, y := q){a)j shall bear a constant ratio, 
n, to the rectangle contained by the latter ordinate and the 
corresponding abscissa. 

By the conditions, we must have 



•/ ( 



q)(x)dx = ^^ ^ , 



n 



and, since this is to hold for all values of a, we may differen- 
tiate with respect to a : hence 

, . (p(a) . (p'(a) 
^^ ^ n n 

cp'ia) __ n—l ^ 
(p{a) a 

and by integration 

lq){a)=z(n—l)la+0. 
Passing from logarithms to numbers, 

cp{a)=: Ca""-^: .' , cp{x) = Cx''-'^ ; 

and the equation of the curve is y z= Cx'^~^. 

Ex.2. Determine such a form for q)(x) that the integral 

w = / -~7^ r shall be independent of a. 

J \/{a — x) 

Put x^=az ; then, since the limits a; =: 0, ic = a, correspond 
to ^ = 0, 2 == 1, 

cp(x)dx r*^ ^aq)(az)dz 



/^ cp[X)ax r* 

o \/(a — x)~J 



^{a — x) J, A^{l—z^ 



INTEGRATION UNDER THE SIGN f. 



451 



By condition, u is to be independent of a : therefore the dif- 
ferential co-efficient of u, with respect to a, must be zero. 

But 

1 



du 
da 



cf (az) , / // N 

Wa + '^""'f ^^'^ _ .« cp{x) + 2xcf\x) . 



V(i-^) 



',\/(a — x) 



and, since this last integral is to be zero for all values of a, 
we must have 

q,'(x) __ _ 1 



q){x) -\- 2xq)^(x) z= : 



(p{x) 



Therefore 



or 



l(p(^x) = — -Ix + 0, 




q)(x)^ 



\/x 




Let A OB be a cycloid, with its vertex downwards ; and let it 
be referred to the axis 
Ox, and the tangent 
through its vertex, as 
co-ordinate axes. Pi 

Then, denoting the 
angle DCF by 0, we V f iT 

have for the co-ordinates 0F= y, OQ = x of the point P, 
X = FF = HL = a -f a cos.^^, 
y = OF-^ All - AE = AR - AD ^ FD 
=z ait — aO -\- a ^'m.O. 
Put =z7t — (f, thou those values of .r and // become 
X = a ~a cos. (jp (\), y -— (uy -j- a sin. iy ('2). 
From (I) wo hnd 
I 



(jT 



COS. 



X . 1 

-, sni.(]r — I -ax — .r 



452 INTEGRAL CALCULUS, 

and thus (2) becomes 



y=aco^r^- \- hax — x"^, 



which is the equation of the cycloid. By differentiation, we 
get 

dx~~y a ' ' 'di^y '^ \dx) ~y X ' 

and, by integration, s = \^Sax. We therefore conclude that 
q){x), in Ex. 2, is the expression for the arc of a cycloid esti- 
mated from the vertex. 

This example is the solution of the problem in mechanics 
for finding the curve down which bodies, starting from dif- 
ferent points, will fall in equal times. 

264. The JEulerian Integral of the First Species 
is an integral of the form 

/>^ 

/ xp-\i — xy-^ dx, 

in which p and q are positive numbers. This is denoted by 

The JEulerian Integral of the Second Species is 

of the form 

r e-^x^'-^dx, 
•^ 

and is denoted by r{n). 

The first species may be put under the two forms 

r y^~^^y_^ , 2 v sin.^^-^ o cos.^^-^ ede, 

by makin2r x = -zr-^ — for the first form, and x =. sin.^^ for the 

1 + y 

second. 



EULERIAN INTEGRALS. 453 

The integral of the first species is a symmetrical function 
of J) and q; for, making x^=^l — y^ we have 

B{p,q) = B{q,p). 
265. Integrating by parts, we have 

JxP{l—xy-^dx 

==- ^lJ^J^j^lCxP-\l-xY-'dx~^CxP{l-^xy-^dx. 
Therefore, taking 1 and for the limits, we have 
B{p + l,q) = P^B{p,q)-P-B{p+\,q): 

.'. B(p + l,q)=-^B{p,q). 
In like manner, 

B{p,q + l)=^^-l-B(i,,q). 

In the integral of the first species, therefore, each of the 
exponents p and q may be diminished by unity. 

200» In the Eulerian integral of tlie second species 

^ 
n must bo positive, oiluM-wiso tlie into^-ral would bo intinito. 
For if 11 be negative, and oipial to ~ p, wo should havo 

and it is pla.iu, that, when .r — oo . the dilVorontial co-oiruMout is 
zero, and theroi'oro the integral is zero; and. when .r — 0. the 



454 INTEGRAL CALCULUS. 

differential co-efficient is infinite, and therefore the integral 
is infinite. 

The integral r{n-\-l) may be made to depend on r(n). 
Eor, integrating by parts, we have 

fe-'^x^dx^z — e-^x"" -\-nJe~''x''-'^dx. 
But e~'^x'^ reduces to zero both when cci= and when a; = oo 
(Ex. 3, Art. 103): therefore 

/e~^x^dx ^n \ e~^x'^~'^dx, 
t/ 

or r{n + 1) = nr{n). 

In Hke manner, 

r{n) =i(n- l)r{n - 1), r(/i - 1) = {n - 2)r{n - 2) ; 
and, if n is entire, we shall have, finally, 

r(2) = r(i), r(i) = C e—dx == i. 

t/ 

Therefore, when n is an entire and positive number, we shall 

have 

r(n) .:= 1.2.3. ..(n — 1); 

and, if 71 is a fraction greater than 1, then the formula 

r{n) ^ {n — l)r{^n — 1) 
enables us to reduce the integral r{ii) to that of r{^), [i de- 
noting a number less than 1. Hence, to compute the value 
r{n), it is sufficient to know the values of this function for 
values of n between and 1. 

267» By putting e~-^ := y, the integral r{n) may be made 
to take another form. Thus, from e~^ = y, we get 

x^^l-, dx =1 ^: 

y y 



EULERIAN INTEGRALS. 455 

268, Melations between the tivo Eulerian Inte- 
grals. Assume the double integral 

and integrate with respect to x : it thus becomes 

Integrating the same double integral with respect to y, it be- 
comes 

therefore 

J {l+yy+^~^^'^^~ r{2y + q)' 



that is 



^0 ^ i{p-\-q) 



Putting - for x in the first member of this last equation, 
we have 

J aP-' ^"^1 a ~ r{p'+q) ' 

Jo ^ ^ I\p + q) 

200» The last formula in the preceding article is a particu- 
lar case of a more general formula, by which may bo expressed, 
in terms of 7^ functions, the nmltiplo integral 

fff...xf''^y'i-^z'--K..^a — x ~~ y — z...)'~'dxd//d:: ... 
extended to all positive values of .r, y, ;:..., which salisly the 
condition .r +// + "••• \ <'• 



456 INTEGRAL CALCULUS, 

Limiting ourselves to three variables^ let 

/a na~x /» a — x—y 

xP-^dx / y'^'^dy \ %^-^i^a — x — y — z) '-^ dz. 

Now, by the last article, 

Multiplying this by y'^~^dy, and integrating with respect to y 
from y z=:{) to y z= a — x, the result is 

(a - ^1^+'-+— nq)nr + s) r{r)r{8) 

-^" ""> r{q + r + 8) ' 

and finally, multiplying this last by xP~'^dx, and integrating 
with respect to x from x := to x =i a, we have 

In this, making a=^l, s = 1, we have 

the limits of integration being any positive values of x, y, z, 
which satisfy the inequality x -\- y -\- z <^ 1. 

Assume Q"=„ (jj = „ (^^J = ^ ; 

then A=^^^-^ \ / / K' \;,^ V d^dridr, 

a^y -^ *^ ^ 

subject to the condition that ^ -{-'/ + C < 1 : therefore 



EULERIAN INTEGRALS. 457 

27 0> By means of Formula 2 of the last article, we can 
find the volume bounded by the co-ordinate planes, and the 
surface having for its equation 

Example. When a = ^ :=: / = 2, and ^ == g = r = 1, the 
surface is that of an ellipsoid of which 2a, 2&, 2c, are the axes. 
Then, by the formula, the volume V of \ of this ellipsoid 
will be 



ahc I V2 



r(1 + i) 



and r f - J =: \/7t; for, let u =z J e~^' c?x, then also 

u=z C e~y\hj, 
*^ 

and ^^-= f" e-'' dx T e'^"' df/ = C C e-"""' -^y"' dxdy. 

Now, r / e~''' ~^" (ix-(/y is obviously \ of the volume the 

equation of wlioso surface is 2 =:: c~-'"^~'''". In terms o^i polar 
co-ordinates, the expression for the same part of the volume is 



r- r zrd-)dr.^ r- re '■"- rd^dr. 



But 






68 



458 INTEGRAL CALCULUS. 

and I dd =z ', • * • 9 J cl^ = j^ ; 

/2 ^ 1 , 

e "^ ax ^^ A^Tt, 

Now, rf-\ — r" e-^ x-^ dx by definition, 

= 2 / e~y'^ dy z=z'±uz=L j^n by putting x^=^y' 
*^ 

1\ ^3 



{ r[ ,. 

a6c ( \2/ ) ahc 
therefore, V =z —^ jt^ — ^= ~ar'^' 

271, Differentiation under the sign f enables us to find 
new integrals from known definite integrals. Thus, 

Example 1. f -^—^ — =Tza~^, 

J Q x^ -\-a 2 

Differentiating each member of this equation n times with 
respect to a, we get 



/ 



, (^2_|_^)«+i -^-2 2 2 2 ^^+^2 

dx __ 1.3. 5. ..(2^ — 1) 7t 



whence J ^ '^^2 j^ ^^n+i~ 2A.6...2n 2a- + ^' 

r°° 1 

Ex. 2. / e-'^'^dx = -• 

*^ a 

After n — 1 differentiations of the two members of this 
equation, it becomes 

f e-«^cc'^-iJa;=:1.2.3...(n— l)a-"; 
that is f e-'^^ic"-! dx =z ^^ (Art. 266). 



DIFFERENTIATION UNDER THE SIGN /. 459 

The last formula holds good when a is replaced by the 
imaginary quantity a + 6 V — 1, in which a is positive ; for 



a-\-h\/—l 



+ C 



^ _ e- icos.hx - V^l sin.5x) ^ 
a + 6 V — 1 
therefore 

f e-ia + b^^)x dx— 7=-- 

and, by differentiating this equality n — 1 times with respect 
to a, we get 

^0 {a + h\/ — If 

272. The formula just found leads to other integrals by 
the separation of real from imaginary quantities. 

Assume a-(-Z>V — l=p (cos. d -}- \/ — 1 sin. ^), in which 

b 



/ -> 7 , ^ 


sin. (9 =r 


p \ a -]-0 , coo.^ Va'' + ^>'^' 


Then 




f g-(a + ^'V-[7x^«-l^^ 





=: f e-"-^{co».hx — s^ — isin.bx)x"-'^dx 

and 

1.2.3...n— 1 _r(n) 1 

(a + 5 V — 1)" "~ P" cos.?i^ -|- \/~^l sin.nd 

= — - (cos.iiO — \^ — 1 sin. }}0) : 



460 INTEGRAL CALCULUS. 

I e~"'^{cos.hx — \/ — 1 sin.hx)x''~^dx 
* * *' 

1= \ ^ (cos. 71^ — \/ — 1 sin. 72^) ; 
an equation which may be separated into the two, 

r 1-77 ^M ■ 

I e~ '^^ x"~'^ siu. oxax z= sm.nd. 



/»00 

/ e~"^ x"~^ COS. bxdx z= 



p 

r(n) 



cos.nd. 



273, Making n=^l in the last formula of the preceding 
article, it becomes 



r _ , , a 

I e "^ COS. oxax = „ , 



- a2 4- 62 • 

therefore, denoting by c a constant less than a, we have 

ada 



But 



c?a / e'""^ cos.hxdx =z I dx \ e~'''^ cos.hxda 

/Qo g — cx g — ax 
cos. hxdx. 
iC 

Again : 

/a ada 1 a^ -{- &^ 

, a''-\-b' ^ 2 c2 + 62 y 

coQ-cx _^-ax 1 a^-\-h'^ 

j cos.oxdxz=-C 

J 



x ' 2 0-^ + &2 

Making & = in the last equation, it becomes 



^ c 



DIFFERENTIATION UNDER THE SIGN f. 461 

a result that may also be obtained by multiplying both mem- 
bers of the equation 

by da, and integrating the result between the limits a and c. 
274. In like manner, from the formula 

/°° . h 

e~"^sin.&x(ix == ., , ,^; 

we get 

I e~'*'^sin.6a:;(i:c =3 / -^ 

c •/ J c CL^ 



+ V 



— tan.~^ = tan.~^ - 

h h 



But 



I e~"-^sin.Z>X(i^ =: I dx I sm.hxe~''^da 

c *^ J Q J c 

/oo g — ex p — ax 
sin. hxdx : 
X 

/ sm. 6a;aa; = tan.- ^ ,^ — tan.- ^ - . 

J X oh 

In this formula, making a = oo , c i= 0, it reduces to 



/°° sin. 6a; , 7t 
ox ~ 2 

whenZ)^;,; whon/;^^^, the sccimuI member boooinos — \^' 
from which it is soon that tlio inteural / '^ • -^ j ^. i-hanu-es 

J X 

abruptly from ^j to —\^, when h, \\\ passiiii;- through zero. 
changes from positive to nogative. 



462 INTEGRAL CALCULUS. 

275. The integral C e-"^' dx = l^7t (Ex. Art. 270) leads 
to / e~ ""^ dx :=. j^ 7t ; for 

«/ 00 

/e~'^\lx=:l e~^^dx-\-\ e~^'^dx. 
—00 J —00 «/ 

Now, if we change x into — x, we have 
e~^' dx z= A^Tt, 

— 00 

And generally, \i f{x) is a function of the even powers of cc, 
that is, such a function that f{x) =/"( — a;), then 

r /{x)dx = 2 r f{x)dx; 

J —00 J 

/oo /^ /» CO 

f{x)dx z= j /{x)dx 4- / /{x)dx. 
— 00 «/ — oo »/ 

But r f{x)dx= r /{—x)dx—J f{x)dx: 

f_j{x)dx=2jy{x)dx. 

In like manner, it may be shown that i^ f{x) is a function 
of the odd powers of cc, that is, if /(— x) — —f(x), we should 
have 

r /{x)dx = 0. 

t/ 00 

/» 00 

j376. In the integral I e-^'c?a?i= >v^;t, putting x^a for 
cc, we have 

»/ -00 va 

which, by n differentiations with respect to a, becomes 



DEFINITE INTEGRALS. 4G3 

In this, making a = 1, we have 

277* Changing x into x -{-a in the formula 

of the preceding article, we get 

r e-^^+^^'dx — ^7t; 

J — oo 

that is, e-«' r" e-^'-2«^cZa; == -</;?; 

J —CO 

But r e-^'-2«^c?a;= r e-^'-2«^c?^ + f^e-^' "^^^cZx, 

«/ —00 J — oo «/ 

and r e-^'-2«^(i:;c= r*e-^'+2ax(ia; 
by changing x into — a: ; 

t/ —00 .Jo «/ 

J 

whence 

»/ 
In this equation, replace a by a>/ — I : then, since 

we have 

/^ _ a X 

t5 -^ COS. 2(U'(7.6'i= ;^o-«"\At. 



464 INTEGRAL CALCULUS. 

This example is another instaDce in which the value of a 
definite integral is found by passing from real to imaginary 
quantities. 

e~^' COS. 2axdx may be 



found consists in differentiating with respect to a and subse- 
quent integration : thus, put 

e~^'cos. 2axdx; 



then 

-— ^ — / sin. 1axe~^^ 2xdx = / sin. 2ax, d,e~^^, 
da J J i) 

Integrating by parts, and observing, that, at the limits, 

sin. 2acce~^" is zero, we have 

-Jf — _ r e-^^ COS. 2aic;2ac?2J = — 2au : 
da J 

da 

da ^^ 2oc. 
u 

But, regarding ^i as a function of a, we have 
du 

da y^ da 

Integrating with respect to a, we get 

by making e"^ =: C. To determine O, make a =: ; then 

/^ 2 1 

e^-^' dx = ~ \/7t = G: 
2 

therefore 

/<» ^ 1 -> 

e~ ^"^ COS. 2aa;c?jc = - e-"' \/7r. 
2 



SECTION X. 

ELLIPTIC FUNCTIONS. 

279, Blliptic Functions or Elliptic Integrals is 

the name given to the following integrals : — 

/e do 
=: F(C, d). 
Vl— c'^sin.-^^ 

9 



Second order. j \/l — c'-^sin.^^ dd = F(c, 6), 

; — = nic, a, 0). 

o(l+asin.2^)Vl-c2sin;^^ 

The constant c is called the modulus of the function, and is 

supposed less than unity; the constant a, which appears in the 

third function, is called the parameter ; and the variable d is 

called the amplitude of the function. The function is said to 

be complete when the limits of the amplitude are and '-. 

The integral of the second order expresses the length of the 
arc of an ellipse estimated from the vertex of the conjugate 
axis (Art. 240) ; the semi-transverse axis being unity, and the 
eccentricity of the ellipse tlie modulus of the integral. From 
this fact, and from the relations which exist between the sev- 
eral functions, the term elliptic functions has been derived. 
Our limits permit us to investigate but a few propositions 
relating to such functions. 

280, Putting X for sin. ^, the integral of the iirst order 
becomes /* "^ dx 

fiS) 405 



466 INTEGRAL CALCULUS, 

In like manner, for another value of x denoted by x-^, we 

have 

r^^ dxi 







•^ Vl - 


-< 


VI 


-c'x\ 




Now assume the relation 












dx 


_|_ 






dxi 




Vi" 


-x's/\- 


-c'x'' 


Vl 


— X 


;Vi_ 


c-'xl 



= (1). 

Multiply through by the product of the denominators, divide 
\)j \ — c'^x^, and integrate ; then 






c^x^x 



z= constant. 
Integrating the first term by parts, we get 

r\/l-xlVl-c-xl ^ x^/\-x\s/l-c^x\ 

I r, dx= ^ 5 

•^ l—c^x-'xl l — c^x^x\ 

, r {^^-c'){l-^c'x'x\)-2cx'-2c'x\ dx, 

~j~ f XX J — =^=^^==: 

•^ {l~^c'x''x\Y \^l-x\s/l-c^x\ 

1 — c^x^x^ 
In this result, interchanging x and cci, we have the second 
term. Adding results, observing that by (1) the terms of the 
sum Avhich are under the sign j reduce to zero, we find 

ccV 1 — x\ V 1 — c'-x^ -\- x^s/\ — x'^ s/l — c'^x^ 

'- ^^-, = const. (2). 

1 — c^x^x 

Eq. 1 expresses the condition that the variables x and Xi are 

so related that the sum of the integrals 

c^ dx n^i dxi 

shall be constant. 



ELLIPTIC FUNCTIONS. 467 

Put r "^ zzza, x^ S{a), 

J VI — x^ vl — c^x^ 

Vr^^^ ^ (7(a), \/l-c''x''^B{a)', 
also 

Vl — cc'J == (7((3'), Vnr^ ^ ^(p/). 
Then, by Eq. 1, we have 

a -{- ^=1 constant = 7. 
It is also seen from (1) that the constant 7 is the value of 
Xi when x — Oj and, further, when a = 0, we have 
x = 0, ^ = r, x, = Sir) = S(a + « : 
therefore, by making the proper substitutions in (2), it be- 
comes 

S(a + ^) = S{a)C{[i)B ( P)-\-Si^ )C{a)B{a) 
1 -c''\jS(a)\' \S(^)\' 
which is the fundamental formula as given by Euler in tlie 
theory of elliptic functions. 

281, Suppose the variables (9, /9i, to be connected by the 
equation 
r^ do r"' — ^^A — f^ ^^ 

or F{c,0)-\-F{c,0,) = F^c,fi), 

in which // is a constant. l[ 0, 0^, be regarded as functions of 
a third variable t, and (1) be diirorontiatod with respect to the 
latter variable, we have 



/.V d). 



dt , dt ^^^ , ^ 



468 INTEGRAL CALCULUS. 

Since the new variable t is arbitrary, let us assume 

f^ = ^{\-cHm?0) (3); 
whence, from (2), 

dt 
Squaring (3) and (4), and differentiating, we get 

d'-d , . ^ ^ d'^d 

^TT == — c sin.(9cos./9, -y— 
dt' ' di' 



1- _^(l_c2sin.2^i) (4). 



7 = — c^ sin. (9 COS. ^, —r~ = — c^sin.^icos./^i 



.•. -^zt-^— — c^(sm./9cos.^=b sm.^icos.^i), 

72 2 

or --(/^zfc^J^z: -~(sin.29±:sin.2^i) (5). 

Put (9 4" (^1 == g), and d — d^^\p\ then 

2^==g) + i/., 2^1 = 9-1/;, 
sin. 2^ = sin. g) cos. ip + cos. g) sin. i/;, 
sin. 2^1 = sin. g; cos. -if* — cos. g) sin. i/; : 
therefore, from (5), we have 

--^ = — c^sm. g) cos. liJ, -rpr = — c^ sm. li^ cos. g). 
We also have 



— c 



c/i c^i^ \dt \dt / 2 



., /cop. 2^ cos. 2^ A „ . 

^= c ( ) =: — c^ sm. g) sm. li; 

d^cp d'^'ip 

dp ^ , W ^. 

= cot. g. 



-') 



c^qp c?i/^ ' dcp dw 

dt dt dt dt 



ELLIPTIC FUNCTIONS. 469 

- c? , . ^ dec d ^ dip dt'^ 

But j^Um. rp = cot. ^^, j^l ^ =-^: 

dt 

d /, dq\ d ^ , d f ^ dw\ d , . 

- U —-=--- 6 Sin. ip, --{ I —-] — — I sm. en. 
dt\ dtj dt ^' dt\ dt) dt ^ 

Whence 

I ^ :=^l sin. 1/; 4" ^> ^ 77 =^ ^ sin. g) + C^i ; 

or by putting C^=IA, Ci^=lA^j and passing from logarithms 
to numbers, 

dx . . dip . . ,„. 

-^^- = ^sm.v-, -rf-^=^.«m.^ (6): 

C?ii; , d(p 

^cos. 1/^ = ^1 cos. g) + (7 (7). 
From Eq. 1 wo see that F{c, 0) = F{c, it) when 6^ = 0: 
therefore we then have d =z fiz= q, z= u^, and (7) then becomes 
{A — Ai) cos. ft = ; and therefore 

^ cos. {0 — 6.^) — .^1 COS. {d + 0^)z=z (A - A^) COS. ^i ; 
whence, by developing cos. (.9 — (9i), cos. (^J -f- ^i), and re- 
ducing, 

{A — A{) cos. Ocos.d^ + (.1 -f Ji)sin. ^sin. ^^1 

= (.i-.-/i)cos.u (8). 
Now, 

dv d) dl, , , o . ., X 

i = ,rt + -,// = Vi I - - --'^^ - N'l ' - ^-^ -"-^ '>.\ 

du> 

Substitute these vahies in [(')), aiul m:iko 0^ -^ 0; then 
^/(l — (.'- sin.-/0 — 1 == v/siii. N. 
\^(l — c- sin.-u') -f 1 -j= ^/j sin. /^ 



470 INTEGRAL CALCULUS. 

From these equations, getting the values oi A-\- A^,A — A^, 
and substituting in (8), we get, finally, 

COS. ^ COS. (9 1 — sin. ^ sin. (9 1-^/(1 — c^ sin.^/w) =i cos. ;w (9). 

This relation, by an easy transformation, may be made to 
take the form 

cos. 1= cos di cos. fi + sin. di sin. ^u ^'(l — c^ sin.^ 0) (10). 

Eqs. (9) and (10) express the connection which exists be- 
tween the variables in two elliptic functions of the first order 
which have a common modulus. 

282, Let F{c, 6), F(c, d^), be two elliptic functions in which 
c, Ci, and (9, (9i, are connected by the equations 

2 4c /ix X . sin. 2^1 .^. 

It is proposed to prove that 

Differentiate Eq. 2, regarding d^ as the independent variable ; 

then 

1 dd __ 2(l + ccos. 2^i) 
cosJd dfi~ (c + cos. 2^i)"^ 

From (2) we also get 

(C + COS.290-- . 
'''''• l + 2ccos.2^i + c-^' 

d^ _^ 2(l + ccos. 2^i) 
^i""l + 2ccos. 2^1+c'-* 

Also, from the same equation, we get 

c^sin.^S^i 



1 — c2sin.2^z=l- 



l + 2ccos.2^i + c'^ 

_ 1 + 2c cos. 2^1 + c^ cos.^29i ^ 
"~ 1 + 2CC0S. 2^1 + c"^ * 



•'• J ^(l 



ELLIPTIC FUNCTIONS. 471 

dd 



_ r 2(1 +c COS. 2^1) v/( l + 2ccos.2^i + c^) 
~J l + 2ccos.2^i4-c2 l+ccos.26'i ^ 



=^/^ 



V(l + 2ccos.2/9i4-c2) 
2 1 dd^ 



''""■ -i^--'- 



J 



4c 2 

But the last integral, when ^^ ^1 7 becomes 

(1 + <^) 



1^ r ^^i 'z_ 

r+cJ V(l-<sin.^^O~l + ^ ^'^ 

If we suppose ^j = - , d = it, then 

7r" 



r+^^i^.'2]=^(^'") = -n^'2 



283, Having shown (Art. 281) that there exists, between 
the variables of two elliptic functions of the first order having 
a common modulus, the relation 

COS. (9cos. ^1 — sin. ^sin. ^1 ^/(l — c- sin. u) = co^. u [\), 

then, between the corresponding functions of the second 
order, there exists the relation 

E{c, 0) + F{c, 0^) — L\c, fi) — c- sin. sin. n ^ sin. u. 
From the equation between tlie aniphtudes 0, 0^, 0^, may 
be considered as a function o( : that is. we mav assume 



472 INTEGRAL CALCULUS. 

and differentiate, thus getting 

v/(l - c^sin.^^) + v/(l - c^sin.^^oj =f'^^)' 

By Eq. 10; Art. 281, the first member of this equation may 
be put under the form 

COS. d — COS. di COS. jLt COS. ^1 — COS. COS. ft ddy 

sin. <?i sin.|ti ' sin. ^ sin. ^u dd 

d{^m}d -\- sin.^/9i + 2 cos. ^ cos. ^i cos.// 1 



do 2sin.^sin.^isin./i 

But putting Eq. 1 under the form 

COS. ^cos. d^ — cos. (U = -^^(1 — c^ sin.^iu) sin. d sin. d^j 
and squaring, we get 

COS.^^ + COS.^^i + COS.'V — 2 COS. d COS. di COS. II 

= (1 — c^sin.^iw)sin.^(9sin.^<?P 
Adding cos.^^i cos.V to both sides of this equation, transpos- 
ing, and reducing by the relation cos.^ ^=1 — sin.^, we find 

sin.^ d -\- sin.^ /9i + 2 cos. d cos. di cos. ^i 

=r 1 -f cos.^ it* + c^ sin.^ d sin.2 d^ sin.^ /*, 

-^ (1 + C0S.2 /i -\- c^ sin.^ d sin.^ ^^ sin.^ ^i) 



2 sin. ^sin. ^1 sin. /i 



_ ni 



c sm. /* 



(i(sin. ^ sin. O^) 



do 

J-\0)=c^ sin., ^^ 



d{sm. ^sin. di) 



and therefore, by integration, 

/((9) = c^ sin. 6 sin. <9i sin. ^. 



\ 



So 



